Cadets, troops and veterans received the traditional Welsh symbol, which had been wrapped in a red thread and dipped in gold.
As Colonel-in-Chief she told the regiment that it was a "great pleasure" to celebrate St David's Day with them.
She also said soldiers are now more engaging with her - suggesting that she has become "less frightening" with age.
Dressed in a heather wool tweed coat by Karl Ludwig, the Queen stood on the dais of the parade square at Lucknow Barracks in Tidworth, Wiltshire, to take the royal salute before the national anthem was played.
She inspected the parade while sat in the back passenger seat of a Range Rover and later met two regimental goats, Shenkin and Llywelyn.
"I am also delighted to be able to present leeks to representatives of the regiment, from cadets to comrades, and to meet the new mascot, Llywelyn," she said.
"The British Army, perhaps more than any in the world, has always lived through the regiment and the regimental tradition.
"In the hour of battle it has repeatedly relied on these bonds, on the pride and comradeship of men who would sooner die than betray the traditions of their corps, or be unworthy of the men of old who fought before them under their colours.
"This is reflected in your regimental motto, Death rather than Dishonour.
"I see that same pride and comradeship in parade before us today and I am certain that the regiment is in good hands.
"I wish you all good fortune for the future."
The Royal Welsh is one of the lead armoured infantry battalions of the British Army and was formed on St David's Day, 1 March, in 2006.
The Queen has been associated with The Royal Welsh and its former regiments, The Royal Welch Fusiliers and The Royal Regiment of Wales, since her coronation.
The Queen has presented leeks to The Royal Welsh regiment to mark St David's Day this week.