WASHINGTON — House Republican leaders, poised for a celebratory takeover of Congress next week, instead found themselves scrambling Tuesday to defuse a racially charged controversy over one lawmaker’s speech a decade ago to a white supremacist group.

After staying silent Monday when news broke that Representative Steve Scalise of Louisiana, the third-ranking House Republican, had appeared before a group called the European-American Unity and Rights Organization, Speaker John A. Boehner and the majority leader, Representative Kevin McCarthy, on Tuesday afternoon issued coordinated statements of support for Mr. Scalise, who acknowledged “regret” and said he had made a mistake.

“More than a decade ago, Representative Scalise made an error in judgment, and he was right to acknowledge it was wrong and inappropriate,” Mr. Boehner said, adding, “He has my full confidence as our whip, and he will continue to do great and important work for all Americans.”

The controversy erupted as Republicans were making a renewed effort to reach out to black voters. It threatened to cloud their agenda after capturing control of the Senate and adding to their House majority in last month’s election. Republicans anticipated using their new power to focus on economic growth and potentially find areas of common ground with President Obama, both elements of a broader push to demonstrate that the party can govern at a time when lawmaking in Washington has all but come to a halt.