Kerry says suggestions of a "parallel government" in Afghanistan are a grave concern.
Afghan official released preliminary election results Monday showing former finance minister Ashraf Ghani Ahmadzai in the lead for the presidency.
But millions of ballots are being checked for fraud and a spokesman for Ahmadzai's opponent, Abdullah Abdullah, has rejected the results and called releasing them a "coup."
Kerry says he expects Afghan electoral institutions to conduct a full review of all reasonable allegations of irregularities.
But he says there is no justification for violence or threats of extralegal action.
He called on leaders to maintain calm.