Washington (CNN) President Donald Trump continued to stand behind his Supreme Court nominee Brett Kavanaugh on Tuesday, even as senators grapple with how to move forward after a woman has accused the judge of a decades-old sexual assault.

"This is is not a man who deserves this," Trump said, adding that the allegations "should've been brought up long ago."

Kavanaugh has denied the allegations.

Trump also criticized Democrats and California Sen. Dianne Feinstein, the top Democrat on the Judiciary Committee, for not bringing up the letter during what he described as a "long, long meeting."

"Why didn't she bring it up then? Because they (Democrats) obstruct and because they resist - that's the name of their campaign against me. They just resist and they just obstruct," he said.

That said, Trump echoed that he and Republicans believe "we should go through a process."

"There shouldn't be a doubt," he said.

"Hopefully the woman will come forward, state her case. He will state his case before members of the US Senate," he said.

Senators "will look at what she had to say from 36 years ago ... and we will see what happens."

Trump reiterated that he feels "terribly" for Kavanaugh and his wife, as well as his "beautiful young daughters."

Later on Tuesday night, Trump tweeted he hoped GOP voters are watching what he called the "Democrats Playbook."

"The Supreme Court is one of the main reasons I got elected President. I hope Republican Voters, and others, are watching, and studying, the Democrats Playbook," Trump tweeted.

The Supreme Court is one of the main reasons I got elected President. I hope Republican Voters, and others, are watching, and studying, the Democrats Playbook. — Donald J. Trump (@realDonaldTrump) September 19, 2018

FBI role

Trump said earlier in the day that he does not believe the FBI should delve any further into the decades-old sexual assault allegation leveled against Kavanaugh , claiming the FBI does not want to be involved.

"I don't think the FBI really should be involved because they don't want to be involved," Trump said.

Trump's comments came as Senate Democrats ramped up calls for the White House to direct the FBI to reopen Kavanaugh's background investigation before any hearings on the allegation of sexual assault leveled over the weekend against the Supreme Court nominated judge can proceed.

The Senate Judiciary Committee has invited both Kavanaugh and Christine Blasey Ford, his accuser, to testify before the committee on Monday.

A Justice Department spokesperson said in a statement on Monday that the FBI forwarded the letter it received last week from Feinstein, which described the initially confidential allegation of sexual assault against Kavanaugh, to the White House Counsel's Office.

"The FBI does not make any judgment about the credibility or significance of any allegation," the spokesperson said, pointing to a 2010 memorandum of understanding between then-Attorney General Eric Holder and White House Counsel Bob Bauer. "Consistent with the memorandum of understanding, the FBI forwarded this letter to the White House Counsel's Office. The allegation does not involve any potential federal crime. The FBI's role in such matters is to provide information for the use of the decision makers."

Trump's own history

Trump has previously voiced suspicion about the #MeToo movement in private, complaining that allegations made decades later can ruin a man's life, people familiar with those conversations say. He has questioned why women wait so long to come forward if they are telling the truth.

During the 2016 campaign, at least 15 women accused Trump of misbehavior ranging from sexual harassment and sexual assault to lewd behavior around women. They came forward in the wake of a 2005 "Access Hollywood" tape that was released in October 2016 in which he is caught saying on a hot mic: "And when you're a star, they let you do it. You can do anything ... Grab them by the p****. You can do anything."

But the White House -- through press secretary Sarah Sanders and others -- has dismissed all the allegations against him as old news that had been litigated during the campaign.

When his staff secretary Rob Porter was accused by two ex-wives of spousal abuse, Trump expressed sorrow that a promising young aide's life had been ruined, even as he recognized that Porter could no longer work at the White House.