Will that increase my website rank for the US in Google?
Yes, because US people do better CTR on SERP and more conversions on your own site.
imagine: your site offers a service for US residents. They are very interesting in it, and, if they see your site's snippet in SERP, they click it wildly. But your site isn't limited to visitors from US. And visitors from i.e. India, looking for such service in google.com, see in SERP your site's snippet wildly too. But then, if they come to your site, the see, the service is only for US residents, and they bounce. Google realizes the big amount of bouncers and thinks, aha, some people bounce, do they probably mean, this site as a kind of shit...? High bounce rate will cause lowering of relevancy.
imagine the same: your site offers a service for US residents. You say it definitely in the site's description, so this information appears in your site's SERP snippet. Visitors, form, i.e. India, are looking for such service, but they don't click your site in SERP, because they read "aha, this service is only for us residents". On this way your site gets lower SERP CTR and, following lower relevancy.
Your site offers a service for US residents. You say it definitely in the site's description, so this information appears in your site's SERP snippet. But additionally you limit the audience of your site to US. Google then shows your site only to US residents - the amount of visitors ramains pretty the same, but on this way your site gets much higher SERP CTR and much lower bounce rate. Both will result in higher relevancy.