Democrats on Capitol Hill gear up for fight over Trump's Supreme Court pick

Battle over Trump's Supreme Court pick

Last Updated Jul 9, 2018 9:56 PM EDT

WASHINGTON -- President Trump Monday night announced Judge Brett Kavanaugh is his choice for the Supreme Court. Now, the battle for confirmation will begin, since Republicans hold a narrow one-seat majority in the Senate.

Before the announcement was made, Texas Republican Sen. John Cornyn voiced his support for the president's short list.

"I honestly think that if it's one of the four, I think that they will be relatively easily confirmed," Cornyn said.

But even as Republicans celebrate, they know a bitter fight lies ahead. Democratic leader Sen. Chuck Schumer set the tone on Monday, warning women's rights are at stake.

"That's what we're up against here," Schumer said. "That's why America is on tenterhooks, so worried about any choice from this list."

Pennsylvania Democratic Sen. Bob Casey, up for reelection in November, announced on Monday that he will vote against the president's nominee no matter who it is, and called the selection process "corrupt."

"The Democrats are not going be happy with anybody," said Utah Republican Sen. Orrin Hatch.

He was rooting for Judge Amy Coney Barrett, whose conservative views were expected to draw the most ire from Democrats.

"She's really outstanding," Hatch said. "There's no question about it and I know the Democrats would go into orbit over it, make it a very tough set of hearings but she can handle it."

Trump adviser: Supreme Court confirmation will be "a big battle"
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    Nancy Cordes is CBS News' congressional correspondent.