"I just think running a third party doesn't feel right," he said in an interview with CNN's Anderson Cooper. "We're not a third party kind of a country."
The Ohio governor said he had received a call asking him to consider a third-party run but that such a path would not be "constructive" or "appropriate."
Kasich also declined throw support behind Trump, telling Cooper he doesn't know if he can ultimately endorse him. "I don't like when he's attacking and putting people down," he said. If he were to endorse Trump, he told Cooper, his wife and daughters would question why he did it.
Kasich said he will attend the convention, which takes place in his home state of Ohio this year, regardless of whether or not he supports Trump. But he emphasized, as he had repeatedly on the campaign trail, that he is not interested in the Vice Presidency.
Kasich suspended his presidential campaign earlier this month, the day after the Indiana primary. He also told Cooper that he thought RNC Chairman Reince Preibus' tweet proclaiming that Donald Trump would be the nominee after the real estate mogul won Indiana and Cruz suspended his campaign "completely inappropriate."
"I'm still there," he said in reference to Priebus' tweet. "I'm not happy about it."