We've had numerous queries about the accuracy of Scorsese's film over the years, but this specific question has been answered by u/washboardsam on the basis of a conversation they had with the film's historical consultant, Tyler Anbinder, a George Washington University historian and author of a book on the Five Points district in which most of the film is set.

You can read that answer here. Anbinder was sceptical because service required citizenship, which was not available immediately on landing, and there was no recruiting station at the point where Irish immigrant ships landed.

It's worth noting that some of the older threads archived here don't meet our current standards and are enthusiastic about the historical accuracy of the remainder of the film – "Yes, Hell-Cat Maggie totally had a jar filled with the pickled ears she had bitten off men in fights" – but be aware that much (including the Hell-Cat Maggie elements) is based on Herbert Asbury's well known Gangs of New York, an elaborated retelling of New York underworld lore that is not, to put it charitably, a fully reliable source. I covered the accuracy, or otherwise, of the film in more detail a couple of months ago in another thread, here.