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John Kasich plans to address his "deep concerns" with Donald Trump

John Kasich on Tuesday said he will soon weigh in on the "deep concerns" he has with his rival campaigns, especially Donald Trump's.

"I will be, however, forced going forward to talk about some of the deep concerns I have about the way this campaign has been run by some others -- by one other, in particular," Kasich told reporters. "But today is not the day to do that."

During the informal presser, he referred to a TV ad from anti-Trump Our Principles PAC that shows women reading Trump's insults at women.

"I just saw a commercial, I guess it was last night of these comments that were made about women," Kasich said. "I have two daughters. They see this stuff. What do you think they think? We'll have more to say about that."

Kasich said there have been moments recently that have been "deeply disturbing" and said he doesn't intend to go "rolling around in the mud," even if he makes it to the general election.

The Ohio governor had just cast his ballot for himself at a polling place in Westerville, Ohio, where he was accompanied by only a few campaign staffers as he was met with a sea of local and national press, cameras in tow. After he voted, he quipped that his 16-year-old daughters would be eligible to vote for his re-election.

The governor predicted he would win his state Tuesday in its critical winner-take-all primary. Recent polls have shown Kasich in a virtual tie with Trump in Ohio.

On CBS This Morning on Tuesday, Kasich touted his "great ground game" and rejected recent surveys that have signaled the neck and neck race.

And on Fox News' "Fox and Friends," he shot down the idea of teaming up with the GOP frontrunner.

"There's no way I would team up with Donald Trump. No way. Forget it," he said. "I'm going to be the nominee because we're going to win Ohio. We're going to move across the country with the positive attitude of bringing people together."

After his press availability, Kasich stepped into his SUV's passenger's side, rolled down the window and thanked the press for their "hard work."

"I appreciate the hard work, the travel, I appreciate it," he said. "It's a lot harder for you than it is for me."

CBS News' Erica Brown contributed to this story.