Is Captain America still a soldier of the US military? If he is, is he required to be a US soldier in order to retain his title of Captain America? Also, is he eligible for further rank promotions?
He's been working for SHIELD, so probably he's not an active member of the military, maybe he's on active reserve, but, due to it's specific abilities I'm quite sure that he's beyond "ordinary military" personnel.
He doesn't need to remain in the military to maintain his title. Normally you only lose your military grade when you're expelled from it, not when you license, retire or enter passive reserve.
He shouldn't earn new promotions, firstly because he isn't on active duty under US Army. He could gain promotions into SHIELD ranks (as he's working for them) but to earn promotions into the military he should be on active duty.
Also, as it's perfectly stated in the answer suggested on comments. Captain is the highest rank for operatives. Ranks beyond it usually comes with different responsibilities, more related to intendency, logistics, strategy and planning. Captain America is a field operative which excels on a very different area than those where a promotion would position him.