I've redacted the transcript to just highlight the important bits.
The short answer is that neither Amy or River are especially keen on the idea of leaving Rory on his own in the past.
Both of them are (rightly) convinced that if she lets the Angel touch her, she'll rejoin her husband.
Although the Doctor is sad about this (as well as worried about her personal safety since he's unsure
where/when she'll turn up), he has no intention of physically stopping her from making her own decision.
The Angel standing behind RORY touches him and he disappears.
......
DOCTOR:
I'm sorry, Amelia.
I'm so, so sorry.
AMY:
No.
No, we can just go and get him in the TARDIS.
One more paradox.
DOCTOR:
Would rip New York apart
and I -- ...
With shuddering breaths, AMY walks towards the Angel.
DOCTOR:
Amy?
What are you doing? ...
AMY:
The Angel, would it send me back to the same time, to him?
DOCTOR:
I don't know.
Nobody knows.
AMY: (takes a step closer)
But it's my best shot, yeah?
AMY:
It'll be fine.
I know it will.
I'll -- I'll be with him, like I should be.
Me and Rory, together.
Melody.
(holds her hand out behind her) ...
AMY: (sobs) Raggedy man, (turns to the DOCTOR) goodbye.
(disappears)
scifi.stackexchange.com/users/20774/Valorum
Why didn't The Doctor just stare at the Angel and tell River to get Amy inside the TARDIS to save her?
Good question.
While I personally consider the curmudgeon's answer (i.e., "the writers messed up") to be the best one to cover that entire episode, there are a few intriguing possibilities.
After time suffering so many pardoxes, the Doctor was becoming essentially "allergic" to the ponds.
Similar to how he cannot bear to look at Captain Jack Harkness, the repeated "wrongness" that Amy carries with her was, by this point, more than he could stand and still think clearly.
(This also neatly answers "why didn't he just walk to New York".
It's not impossible for, say, River to do so -- it's something that the Doctor simply cannot physiologically endure.)
The Doctor knows that Amy loves Rory, and that he cannot stand in the way of his friend's last chance to spend her life with her one true love.
(See @Richard's answer for more on this.)
Clara distracted him, as part of her appearing throughout his timeline to help steer him towards her continuity.
(If the Doctor doesn't lose Amy, he doesn't sulk.
If he doesn't sulk, he doesn't meet the GI and Clara.)
And without Clara, he doesn't save Galifrey.
The Doctor is lying, and has known for some time that this is where he leaves Amy and Rory.
He's not crying out of fear; he's crying because, once she goes to NY without him, she's no longer his companion.
scifi.stackexchange.com/users/16726/DougM
It's easier than that - Amy stepped between the angel and the doctor.
He couldn't see the angel.
She blocked his view with her body.
But River knew that it was the right thing to do so she blinked - for Amy.
River was willing to give up both of her parents just so they could spend their lives together.
scifi.stackexchange.com/users/29197/user29197
Perhaps it's a also that The Doctor read the name of the last chapter of River's book, which I believe was "Amy's Goodbye".
He clearly believes that everything in the book must happen, as can be see by how River must break her own wrist earlier in the episode to free herself.
Now that he sees how Amy is definitely saying goodbye to him, wouldn't he realize that it all fits in to place and not try to stop her?
He is sad but realizes it was already destined to happen, according to River's book.
scifi.stackexchange.com/users/24232/tethernova
The short answer was that the Doctor was aware that Amy and Rory in the past had become a fixed point in time.
This happens when he reads the chapter name
"Amelia's Last Farewell" So he could do nothing to change that except play it out.
Although he tried to prevent the fixed point by refusing to break Rivers wrist to free her.
River then breaks her own wrist, thus fulfilling the storyline of the book.
There was also a fan theory for that season that the episodes were running in reverse order in relation to the doctors timeline.
So a lot of the actions the doctor takes in earlier episodes is because of the outcome in Manhattan.
This is certainly the case with River's timeline and she was the one who also told her to be with Rory.
scifi.stackexchange.com/users/12262/Simon O'Doherty
At the end of The Angels Take Manhattan, we see a surviving Weeping Angel taking Rory away right outside the TARDIS. Then Amy lets herself be touched by the Angel to join Rory in the past.  What troubles me is that both The Doctor and River were standing right there and witnessed the whole thing. Why didn't The Doctor just stare at the Angel and tell River to get Amy inside the TARDIS to save her?
Why didn't The Doctor just stare at the angel to save Amy?
scifi.stackexchange.com
<doctor-who>
scifi.stackexchange.com/questions/48843
scifi.stackexchange.com/users/21184/tilley31
The Doctor knows that Amy loves Rory, and that he cannot stand in the way of true love.  He clearly believes that everything in River's book must happen, as can be seen by how River had to break her own wrist earlier in the episode. Amy stepped in the way of the angel and the doctor thus preventing this from happening. Also, the doctor couldn’t break the fixed point in time and had to play out the reality of what he already knew and thus could not save her.
Amy stepped in the way of the angel and the doctor. The doctor couldn’t break the fixed point in time and had to play out the reality of what he already knew which meant he couldn’t save her.