A campaign-finance watchdog group filed a complaint against Health and Human Services Secretary Tom Price on Friday, alleging he improperly used his congressional campaign fund to promote his confirmation to join President Donald Trump’s cabinet.

In its complaint with the Federal Election Commission, the Campaign Legal Center cited a $40,000 payment the former Georgia congressman’s campaign committee made on Jan. 26 to America Rising, a GOP consulting firm. Around the same date, the complaint said, the firm released research...