This is something the comics world has been discussing for a while now, and it seems to be getting ready to come to a head soon. Now, first let me say that technically, Cap is not a Nazi at the moment. However, he has been turned evil and "has been an agent of Hydra from the beginning" (the quotes are important). Hydra is inextricably linked in the minds of many to be a Nazi organization. The current canon origin of Hydra was in ancient Egypt, long before Hitler could have ever been conceived. As Hydra stands now in the comics universe, there are two factions. One is the Red Skull's faction, who is undeniably a Nazi. The other is a more general New World Order group, which Cap is part of.
Now, let's go back to the quotes from earlier. Was Captain America always a member of Hydra? No. Red Skull obtained a cosmic cube and altered reality to make Cap a Hydra agent now. In doing so, he also altered Cap's memories (and those of others involved) to reflect a history of him being Hydra all along.
Last week's Captain America comic clarified things a little further, actually. It showed one of those "memories" wherein Cap says Red Skull betrayed Hydra by connecting it with the Third Reich. However, in the minds of most, Hydra is purely a Nazi organization.
Now that the backstory has been put forth, it's time for some commentary. Yes, the idea of Captain America as a Nazi is unthinkable and goes against everything the character is about. However, let's look at this from a story perspective. Say you're Red Skull, who has been foiled time and again by your worst foe, Captain America. You gain the power to alter reality. Wouldn't it make sense to transform that foe so that he wouldn't be a thorn in your side, but an ally? In fact, he didn't go far enough with the change, as Cap murdered Red Skull in order to fully take over Hydra. Oops. Logically, he should have made Cap a full Nazi, while I still hate the idea.
Now, will Nazi Cap always be on people's minds from now on? Hard to say, though he will be for a while at least. In the end, will he be remembered forever like alcoholic Tony Stark or eventually forgotten like evil, murderous Tony Stark and Teen Tony Stark? Only time and how it is handled will tell.