WASHINGTON — A political group backing President Trump is firing a warning shot at the anti-tax Club for Growth, which opposed the initial White House effort on repealing the Affordable Care Act, with a new ad coinciding with a special congressional election in Georgia on Tuesday.

The spot, by the 45Committee, which was heavily funded by the casino magnate Sheldon Adelson and the wealthy Ricketts family during the presidential campaign, centers on the Club for Growth’s support for Bob Gray, one of the Republican candidates in the 18-person field in the special election in Georgia’s Sixth Congressional District.

But hitting Mr. Gray is a proxy fight in which the 45Committee, one of the biggest outside groups behind Mr. Trump in the 2016 presidential race, is taking a shot at a potential Trump critic.

The Run-Up

The podcast that makes sense of the most delirious stretch of the 2016 campaign.

The spot, airing nationally on Fox News, describes the conservative Club for Growth — one of the best-funded groups on the right — as being responsible for “saving” President Barack Obama’s signature health care law by criticizing the initial deal the White House considered. The spot goes on to accuse the club of “saving” Planned Parenthood as well, and notes that it “attacked” Mr. Trump.

Interactive Feature | Get the Morning Briefing by Email What you need to know to start your day, delivered to your inbox Monday through Friday.

The spot coincides with a direct-mail campaign that the 45Committee sent to about 100,000 homes, criticizing Mr. Gray. The ad buy is roughly $300,000, an official for the 45Committee said.

By airing it on Fox News, there is at least some chance that Mr. Trump, an avid cable news viewer, will see it.

“The Club for Growth has actually spent more money this year against conservatives than the D.N.C., D.S.C.C. and D.C.C.C. combined,” said Brian Baker, the 45Committee president, using the abbreviations for the Democratic National Committee and the Democratic senatorial and congressional campaign committees.

“At the same time, the Club for Growth saved Obamacare from repeal and ensured continued taxpayer funding for Planned Parenthood,” he added. “Now they’re attacking some of the best and brightest young leaders in the conservative movement in an attempt to shift the blame. They’ve gone too far.”

A spokesman for the club declined to comment, saying the group was focused on the Georgia special election.

The vote on Tuesday will decide who should fill the seat vacated by Tom Price, a Republican who left the House to become Mr. Trump’s health and human services secretary.