No, Napoleon Bonaparte was not executed after being exiled as leader of France.
He tried unsuccessfully to officially step down and have his son put on the throne.
But when that idea was rejected, he abdicated voluntarily anyway and was exiled to the island of Elba in the Mediterranean. His wife and only child were sent to live with his father-in-law—the emperor of Austria—and he never saw them again.
He did, however, manage to escape Elba. In 1815, he returned to France, rallied the troops, and marched on to Paris. He quickly tried to make peace with his former enemies but was refused.
When pushed to battle, he was defeated by the British at the battle of Waterloo.
Despite his popularity back home, Britain and its allies exiled him once more—this time for good—to the island of St. Helena in the South Atlantic Ocean, where he died of stomach cancer six years later.