user3237: I'm a 1750 Elo player. I have a very positional, quiet and strategic style of play.
As White I play 1. d4, but I plan to also learn 1. Nf3 soon. As Black against 1. e4 I play the Caro-Kann.
I would like an opening for Black against 1. d4 which meets these two conditions:

It needs to be as much similar to the Caro-Kann as possible (since I already play the Caro-Kann against 1. e4).
On average it must tend to lead to positions which are as much solid, positional, slow, closed, strategic and quiet as possible (I want nothing sharp and tactical).


user8291: To try to minimize tactical complications early in your games against 1. d4 I would recommend you the Classical Slav, (but not the Semi-Slav), although very sharp and tactical positions may arise in every chess game! 

user3495: I agree with a previous answer that it's hard, and maybe pointless, to seek a similar defense to the Caro Kann when facing 1.d4. 
But for your other requirements of positional solidity what about The Old Indian defense? https://en.wikipedia.org/wiki/Old_Indian_Defense
The Old Indian is solid and respectable; it was played by Grandmasters such as Lothar Schmidt. In fact looking at his games now he played it a lot and described it as his "old friend." It's objectively fine; it's just lacks the hair raising tactics of the King's Indian.  

user18809: Some good ideas in the rest of the answers, but I'd like to add that if you like positions in the Panov attack, you should probably try the Queen's Gambit Declind, perhaps the Tarrasch variation. You'll get to plenty of isolated 'd' pawns from there, playing both with and against it.
The Slav Defence appears to look like a Caro-Kann, but more often that not it looks nothing like it unless you allow your opponent to play e4, which is often not a good idea, as you'll probably have a worsened version of the classical Caro-Kann

user19861: The semi-slav is the perfect compliment to the CK however it might be more tactical than what you want. The slav is good but drawish. 
I would recommend playing 1...c6 against everything and working out the transpositions. The Slav is probably fine for what you want but if that opening is too passive the semi-slav gives you a more aggressive game. 
If you don't like that, you have your choice of the QGD (1...d5), various Indian defenses (1...Nf6) or the flexible (1...e6)  
On a side note, the colle for white fits very well with the CK/Slav ideas as black. A rep of colle/slav/ck is pretty simple to learn and reaches similar positions from both sides. It's all built around the same pawn structures. 

user2042: 
It needs to be as much similar to the Caro-Kann as possible (since I already play the Caro-Kann against 1. e4).

Impossible. The problem with this approach is that you have already stopped e4 with ...d5 so White simply can not transpose even if he wished to do so. Your best bet is the Slav defense, as it is very similar to Caro-Kann ( same pawn structure for Black, with the Bf5 bishop ), but even then there are ways for White to deviate into sharp and very different lines.

On average it must tend to lead to positions which are as much solid, positional, slow, closed, strategic and quiet as possible (I want nothing sharp and tactical).

The best candidate fitting into this description is Queen's Gambit Declined. 
Only Black can sharpen the game by transposing into Semi-Slav with Botvinnik system or with some sharp lines in the Slav with ...b5, but if you wish you can always steer the game into solid, positional lines. Then White will have no way to sharpen the game ( neither do you ), and you both will have to battle with dull, positional maneuvers and plans. 
Be warned, it is very hard to play these positions as Black and tactical skill here is practically irrelevant! Most of the time you will have a hard time defending, and you will have to learn how to properly engineer a freeing break. That is not an easy task, just so you know ( I am speaking this from experience )! 
In order to play this opening well with either color you also must know how to play typical positions very very well ( not to mention you need strong endgame knowledge! ), but this might suit your style ( based on your description in second criteria ). 
SUMMARY:
Chess would be too simple if you could play one opening against every opponent's reply. This especially applies for Black! I once too wanted to play the Caro-Kann against everything, but after very thorough study I was forced to realize that it isn't possible.
As for positional opening against 1.d4, your best bet are the dull lines of the Queen's gambit declined. In every other opening White always have a chance to sharpen the game ( always! ), but if you choose this path be ready for a long defense which will usually give you no more than a draw. Still, you will get the solidity and quietness you seek...
If you need further help or clarifications leave a comment.
Best regards.

3237: <p>I'm a 1750 Elo player. I have a very positional, quiet and strategic style of play.</p>

<p>As White I play 1. d4, but I plan to also learn 1. Nf3 soon. As Black against 1. e4 I play the Caro-Kann.</p>

<p>I would like an opening for Black against 1. d4 which meets these two conditions:</p>

<ul>
<li>It needs to be as much similar to the Caro-Kann as possible (since I already play the Caro-Kann against 1. e4).</li>
<li>On average it must tend to lead to positions which are as much solid, positional, slow, closed, strategic and quiet as possible (I want nothing sharp and tactical).</li>
</ul>
 8291: <p>To try to minimize tactical complications early in your games against 1. d4 I would recommend you the Classical Slav, (but not the Semi-Slav), although very sharp and tactical positions may arise in every chess game! </p>
 3495: <p>I agree with a previous answer that it's hard, and maybe pointless, to seek a similar defense to the Caro Kann when facing 1.d4. </p>

<p>But for your other requirements of positional solidity what about The Old Indian defense? <a href="https://en.wikipedia.org/wiki/Old_Indian_Defense" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Old_Indian_Defense</a>
The Old Indian is solid and respectable; it was played by Grandmasters such as Lothar Schmidt. In fact looking at his games now he played it a lot and described it as his "old friend." It's objectively fine; it's just lacks the hair raising tactics of the King's Indian.  </p>
 18809: <p>Some good ideas in the rest of the answers, but I'd like to add that if you like positions in the Panov attack, you should probably try the Queen's Gambit Declind, perhaps the Tarrasch variation. You'll get to plenty of isolated 'd' pawns from there, playing both with and against it.</p>

<p>The Slav Defence appears to look like a Caro-Kann, but more often that not it looks nothing like it unless you allow your opponent to play <strong>e4</strong>, which is often not a good idea, as you'll probably have a worsened version of the classical Caro-Kann</p>
 19861: <p>The semi-slav is the perfect compliment to the CK however it might be more tactical than what you want. The slav is good but drawish. </p>

<p>I would recommend playing 1...c6 against everything and working out the transpositions. The Slav is probably fine for what you want but if that opening is too passive the semi-slav gives you a more aggressive game. </p>

<p>If you don't like that, you have your choice of the QGD (1...d5), various Indian defenses (1...Nf6) or the flexible (1...e6)  </p>

<p>On a side note, the colle for white fits very well with the CK/Slav ideas as black. A rep of colle/slav/ck is pretty simple to learn and reaches similar positions from both sides. It's all built around the same pawn structures. </p>
 2042: <blockquote>
  <p>It needs to be as much similar to the Caro-Kann as possible (since I already play the Caro-Kann against 1. e4).</p>
</blockquote>

<p>Impossible. The problem with this approach is that you have already stopped <code>e4</code> with <code>...d5</code> so White simply can not transpose even if he wished to do so. Your best bet is the Slav defense, as it is very similar to Caro-Kann ( same pawn structure for Black, with the <code>Bf5</code> bishop ), but even then there are ways for White to deviate into sharp and very different lines.</p>

<blockquote>
  <p>On average it must tend to lead to positions which are as much solid, positional, slow, closed, strategic and quiet as possible (I want nothing sharp and tactical).</p>
</blockquote>

<p>The best candidate fitting into this description is <em>Queen's Gambit Declined.</em> </p>

<p>Only Black can sharpen the game by transposing into <em>Semi-Slav with Botvinnik system</em> or with some sharp lines in the Slav with <code>...b5</code>, but if you wish you can always steer the game into solid, positional lines. Then White will have no way to sharpen the game ( neither do you ), and you both will have to battle with dull, positional maneuvers and plans. </p>

<p>Be warned, it is very hard to play these positions as Black and tactical skill here is practically irrelevant! Most of the time you will have a hard time defending, and you will have to learn how to properly engineer a freeing break. That is not an easy task, just so you know ( I am speaking this from experience )! </p>

<p>In order to play this opening well with either color you also must know how to play <em>typical positions</em> very very well ( not to mention you need strong endgame knowledge! ), but this might suit your style ( based on your description in second criteria ). </p>

<h2>SUMMARY:</h2>

<p>Chess would be too simple if you could play one opening against every opponent's reply. This especially applies for Black! I once too wanted to play the Caro-Kann against everything, but after very thorough study I was forced to realize that it isn't possible.</p>

<p>As for positional opening against <code>1.d4</code>, your best bet are the dull lines of the Queen's gambit declined. In every other opening White <em>always</em> have a chance to sharpen the game ( always! ), but if you choose this path be ready for a long defense which will usually give you no more than a draw. Still, you will get the solidity and quietness you seek...</p>

<p>If you need further help or clarifications leave a comment.</p>

<p>Best regards.</p>
