Working hand in hand with the White House, Mr. McConnell has devoted substantial time and energy to installing scores of Trump-nominated judges over the past three years and portrays that as a main achievement of the Republican-led Senate. Democrats are trying to convince voters that a Democratic majority in the Senate would slow the Republican judicial juggernaut or begin to offset the effort should a Democrat win the White House.

Democratic officials and their progressive allies said that while Mr. Schumer might have gone too far with his inflammatory words, they were not all that unhappy that the episode put new attention both on the Louisiana case at hand and the Republican push on the courts over all.

“Donald Trump’s attacks on Senator Schumer are disingenuous and dishonest, as always,” said Marge Baker of People for the American Way, a liberal advocacy group. “The truth is that the electorate will respond vigorously in the fall to efforts to strip people of reproductive rights, and that’s what Republicans are afraid of.”

In his Thursday comments, Mr. Schumer also sought to shift the focus from his own messaging to the issue of abortion and Senate Republicans.

“To the women of America, what we are talking about here, what I am fighting for here, is your right to choose — an issue, of course, Leader McConnell completely ignored in his speech,” he said.

Mr. Schumer said the courts would most likely curtail abortion rights “because Senate Republicans have confirmed nominees they believe will strip away women’s rights and fundamentally change this country, going so far as to deny a duly-elected president the right to pick a Supreme Court justice.”

That comment was a reference to Mr. McConnell’s decision in 2016 to refuse to act on President Barack Obama’s nomination of Judge Merrick B. Garland to the Supreme Court, instead keeping the court seat open until Mr. Trump won the election and took office. The blockade still infuriates Democrats.