"If he was going to shake his hand, he should have asked him about those basic freedoms Mandela was associated with that are denied in Cuba," Rubio told ABC News.
Obama actually did make reference to authoritarian leaders during his eulogy for Mandela.
"There are too many leaders who claim solidarity with Madiba's struggle for freedom, but do not tolerate dissent from their own people," Obama said.
Rubio's response represents the first real criticism Obama has received from a major Republican for the handshake. Former President Jimmy Carter called the handshake "something significant" and expressed hope that it "will be an omen for the future."
This post has been updated.