Probably not for saying so, now if he started indiscriminately bombing other countries on his own whim and made up reasons, the result would be far worse than impeachment.
An impeachable offense is a high crime or misdemeanor. Consistent malfeasance of office could be considered impeachable, the clause is elastic enough for that. But, that's not what would happen. What would happen to the US is just what we did to Saddam Hussein. A coalition of willing countries would be formed. obama and his regime would be given a set time to leave the country. Hopefully for the regime, some other nation would take them in as exiles. If obama did not leave during the specified time the United States would be attacked from all sides, a resistance in the United States and whatever portion of the military that had broken away would help them.
Just what happened in Iraq.