Fresh off his successful Supreme Court battle, President Trump will now try his luck stocking the federal appeals court with conservative jurists — and Democrats are mostly powerless to stop him.

First up is Judge Amul R. Thapar, whom Mr. Trump has tapped to sit on the 6th U.S. Circuit Court of Appeals and who will sit for his confirmation hearing in the Senate on Wednesday.

Progressive activist groups are preparing for a fight, saying Judge Thapar has several controversial rulings on his record.

But Senate Democrats said they have yet to examine Judge Thapar’s nomination and haven’t revealed a strategy for him or for the rest of Mr. Trump’s federal court nominations going forward.

Sen. Christopher A. Coons, Delaware Democrat, pointed to other picks he said were higher priorities for the Senate to tackle.

“I hope we will get to a process of considering nominees for the Justice Department, for the State Department, where in those case where we can quickly come to an agreement, we’ll process them so the government can function,” said Mr. Coons. “And in those cases where there’s reason for sharp disagreement, we’ll continue to stand firm against those who aren’t qualified to serve.”

Mr. Trump has an opportunity to reshape the federal judiciary.

Circuit courts need nominees for 19 vacancies, and district courts are waiting to fill more than 100 vacancies, but the White House has been slow to announce names.

Part of the reason for the slow pace is that the Senate has been consumed with the fight over Justice Neil Gorsuch. The justice was confirmed this month — but only after Democrats mounted a filibuster, forcing Republicans to use the “nuclear option” shortcut to change the rules.

Use of the filibuster won’t be an obstacle for the lower-court picks, either, because Democrats used the nuclear option in 2013 to defang the procedure for all non-Supreme Court nominees.

A committee aide said Democrats will insist that the Senate respect the “blue slip” tradition, which gives senators a chance to approve judicial picks from their home states. If both senators don’t return their blue slips, nominations traditionally are scuttled.

Sen. Dianne Feinstein of California, the ranking Democrat on the Senate Judiciary Committee, has also signaled that she will ask the committee to wait for evaluations by the American Bar Association before nominees get confirmation hearings. The Trump administration has said it won’t wait for an ABA rating before making its nomination, breaking with the practice of the Obama administration.

Progressive groups, which pushed the nuclear showdown over Justice Gorsuch, say they still want Democrats to test Judge Thupar.

“Hopefully, the Senate confirmation hearing tomorrow will illuminate his views and what type of judge he will be on the 6th Circuit,” said Dan Goldberg, legal director for the progressive Alliance for Justice. “We intend, after the hearing, to certainly further educate the full Senate on Judge Thapar’s record.”

Mr. Goldberg said Judge Thupar was on the list of potential nominees that Mr. Trump released last year as a presidential candidate. Democrats said the list was suspect because it was crafted with help from conservative legal advocates.

Mr. Goldberg also said Judge Thapar has issued troubling rulings, including opposing campaign ethics rules for judicial candidates in Kentucky that would have limited their contributions in partisan politics.

Judge Thapar was later reversed in part by the 6th Circuit — the court for which Mr. Trump has nominated him.

Mr. Goldberg also said Judge Thapar rejected a family’s lawsuit against a jailhouse nurse who they claimed had acted indifferently to their family member’s medical needs. The family member died of diabetic ketoacidosis while in jail.

Conservative groups were skeptical of Democrats’ appetite for another bruising fight over judges.

“It’s not Supreme Court stakes at this point, and the Democrats have to go in recognizing that Judge Thapar is someone who will easily be confirmed,” said Carrie Severino, chief counsel at the conservative Judicial Crisis Network.

She said Judge Thapar is a “stellar nominee” and is particularly strong on safeguarding constitutional limits for all branches of government — including the courts.