(CNN) The day after he was nominated to become a Supreme Court justice, Brett Kavanaugh visited Capitol Hill, where his possible future on the high court will be determined.

Kavanaugh and Vice President Mike Pence met Tuesday with Senate Majority Leader Mitch McConnell. The meeting came at almost the same time that Senate Democrats held a rally against Kavanaugh in front of the Supreme Court, an event led by Senate Minority Leader Chuck Schumer and the Democrats on the Senate Judiciary Committee, which will hold Kavanaugh's confirmation hearing in the coming weeks.

Kavanaugh also met later Tuesday with the committee's chairman, Republican Sen. Chuck Grassley of Iowa, for what Grassley described as a "pleasant conversation."

"We're going to have a thorough process. Hopefully it's efficient, we get it done quickly," he said, adding later, "In the end, I think his record will speak for itself."

The pick of Kavanaugh announced Monday night immediately launched the highly polarized fight in the US Senate , testing both sides' ability to unify their ranks just months ahead of the hugely consequential midterm elections.

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