"Everything President Trump said today is simply wrong, end of story," a spokesperson for Kerry told CNN.
"He's wrong about the facts, wrong about the law, and sadly he's been wrong about how to use diplomacy to keep America safe."
Trump asserted earlier Thursday at the White House that Kerry had violated the Logan Act, which bans private citizens from negotiating with foreign governments without permission from the administration.
"John Kerry speaks to them (Iran) a lot, and John Kerry tells them not to call," Trump said. "That’s a violation of the Logan Act, and frankly he should be prosecuted on that."
Kerry played a key role in negotiating the nuclear deal between the US and several world powers with Iran during the Obama administration, a deal Trump pulled out of last year.
The former secretary of State said last year that he had met with Iran's foreign minister "three or four" times since leaving office to discuss the nuclear pact, among other issues.
Violating the Logan Act is a felony, but only two people have ever been charged with doing so and no one has ever been convicted of violating it, according to The Hill.