WASHINGTON -- Ohio Gov. John Kasich said he's not going to run for president as a third-party candidate, dashing hopes he might return to the race after businessman and former reality TV host Donald Trump becomes the Republican Party's presumptive nominee.

The governor told CNN's Anderson Cooper he thought it would be "a silly thing."

"And I don't think it's appropriate," Kasich said. "I just don't think it would be the right thing to do."

He and Texas Sen. Ted Cruz both made way for Trump after Trump's win May 3 in the Indiana GOP presidential primary. Still, some in the party have questioned whether Trump can win against presumed Democratic nominee Hillary Clinton.

Kasich said he hasn't been swayed, though.

"I gave it my best where I am," he told Cooper. "I just think running third party doesn't feel right. I think it's not constructive."

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Still, Kasich has so far declined to endorse Trump, saying he wondered what his daughters would think of him if he did so. And he reiterated to Cooper that he wouldn't consider being Trump's vice presidential candidate, saying his views were simply incompatible with those Trump has promoted.

"Those are two very inconsistent messages, so it would be very hard for me -- unless he were to change all of his views and become a uniter -- for me to get in the middle of this thing," Kasich said. "Because, you know, I'm undecided here about what I'm going to do in this race."

He also warned that Trump could lose Ohio in November if he doesn't change his tone: Republicans, he said, will have a hard time winning the White House if their candidates are "bashing Hispanics, turning off African-Americans" and failing to excite young voters.

A poll released two weeks ago, just a day before that game-changing Indiana primary, found a majority of Ohio voters wanted him to drop out of the Republican race, and of those, most said he was neglecting his gubernatorial duties in the Buckeye State. Also mentioned in the poll, from left- was the expense of his security detail on the campaign trail -- which is paid, essentially, by Ohio taxpayers.

Ohio was the only state Kasich won in his effort to secure the Republican Party presidential nomination. The state's 66 Republican delegates will still be required to vote for Kasich at the Republican National Convention later this year.