Why would the Stolen Valor Act be unconstitutional?

Because someone impersonating a police officer has direct consequences to public safety. Someone lying about their service does not.

Because in theory, our soldiers, sailors, airmen, Marines, fight for all freedoms, not just the ones they like. It's not like a soldier can say, "Okay, I'm fighting for people's right to worship in Lutheran churches but not Anglican churches."

Likewise, the First Amendment protects your right to say anything, whether it's true or false, whether it's philosophically profound or just plain nonsense. Therefore, people have a constitutional right to lie about their military service if they want. In fact, the DD-214 (for those who actually served) has a copy that only lists the time you served, and not the character of your discharge (honorable, general, bad conduct, etc.)

But don't worry: just as the First Amendment protects Jackie Albert Stern or Michael F. O'Brien's right to say they won the Medal of Honor, it also protects your right to call them liars.

Though you bring up a good point about impersonating police officers, the issue involved would appear to be very similar. However, the Supreme Court has always upheld that urgent public safety concerns can outweigh or delay freedom of speech considerations. You know the classic yelling "fire" in the theater scenario, right? You have the right to say that word, but you don't have the right to yell it in order to needlessly panic others.

Dressing up like a cop and acting like you have all the rights and responsibilities of a cop is a very different matter than saying you're a cop while dressed in civilian clothes and supposedly off-duty. It doesn't really endanger anyone if I were to claim that I was in a Marine recon unit and that I killed one of Saddam Hussein's decoys with my own bare hands. It does endanger people if I were to dress up like a cop and arrest some people I don't like.

I only wish to add to Lisa's response that most Medal of Honor impostors don't do it for financial gain, whereas that it is usually the case with a police impostor.

because it just is