Some teams (e.g. Baltimore and New England) do not have this patch on their jerseys but still designate captains.
A response on Patriots.com confirms this and adds that it may have something to do with Belichick's philosophy:
We've asked Belichick about this in the past and haven't really gotten a semblance of an answer. Basically he's said they don't wear the patch because they don't want to. He went on to say that football survived for decades without the captains wearing patches and got along just fine. It's obviously not a requirement but most teams adhere to it.
And ESPN says something similar:
Later, when asked why the Patriots don't wear captains patches, Belichick tossed a little zinger at the NFL. "I guess that's another one of the new traditions in the league. I've seen a lot of football games, there has been football for a long time, and there have been captains with no patches and it seemed like everything was fine. I guess that's another one of our improvements, with the new overtime rules and all that."
All of this points to it being a choice by Belichick not to have the patches on his captain's jerseys.