Many Republicans, who swept to recent electoral victories by vowing to topple the ACA, are urgently seeking to reassure voters they want to save these protections. Mr. Rohrabacher says he is “taking on both parties” in an effort to do so; his challenger says the congressman is “falling all over himself to scrub his records on health care.”

Such fights are leading to sometimes bitter races across the country involving the ACA, enacted in 2010 under President Obama.