No. But NOT because he was impeached. Federal law requires that once someone has served as POTUS, they cannot run for any other elective office.
However, that does not preclude them from being APPOINTED to serve in a variety of unelected capacities. Again, Clinton and his liberal predecessor Carter provide us with examples.
In Clinton's case, he wanted to run for another elected office after his second term ended in 2000, but was prevented by federal law. In Carter's case, look how many times he has been appointed as a "special ambassador" to represent the U.S.' interests to a variety of oppressive dictators whom he loves and gets along with - to the disadvantage of America.
I think Obama could try and run for President again as he has only served one term. If he loses this time he’d still have one term he could run for???
Did I miss the passage of a new law? This law would of interest to the Presidents that have been elected to Congress after serving their terms as POTUS, such as Adams and Johnson.