“They say he was legally removed from office as he had violated the Honduran constitution.”
Can we do that here?
Yes, if the President violates the Constitution, he can be removed. But obviously it's not easy. For instance, if Obama is really a Kenyan/Indonesian who lied under oath about his citizenship, the courts could remove him.
But, as in Honduras, it would probably require the military to side with the courts rather than with the bogus President. Certainly the Justice Department under that crook Holder would side with Obama. Once the legitimacy of the government is called into question, all bets are off until the matter is resolved, which requires popular support and approval of those with the power to prevent it.
Or if Obama violates the Constitution in some other way--as he in fact has done in such matters as the GM takeover--Congress can impeach and the Senate convict him. That's unlikely at this juncture, unless he becomes so widely hated that the Democrats consider him a hopeless liability.