Even so, prominent Republican leaders in New Mexico have somewhat refrained from attacking the broader containment strategies established by Gov. Michelle Lujan Grisham, focusing instead on the economic strain in the state.

“Our main difference with the governor is the inequity,” said Steve Pearce, chairman of the Republican Party in New Mexico and a former candidate for governor who lost to Ms. Lujan Grisham in 2018.

Mr. Pearce said he was focusing criticism on Ms. Lujan Grisham’s decision to allow large stores like Walmart to remain functioning while smaller operations are shut down. “It’s unfair to side with major out-of-state corporations over local businesses,” Mr. Pearce said.

While New Mexico has not experienced so-called reopen protests of the size or intensity as those in other states, some legal challenges to virus-containment measures are emerging from conservative groups, including the National Rifle Association.

“Our governor left abortion clinics open but closed us down,” said Steve Smothermon, the senior pastor of Legacy Church, an Albuquerque megachurch that sued Ms. Lujan Grisham this month over a ban on mass gatherings in places of worship.

A judge rejected Legacy’s request to throw out the ban. Mr. Smothermon is holding Friday night services in Legacy’s parking lot where congregants can tune into services from their cars.

Ms. Lujan Grisham said her previous experience as state health secretary and director of New Mexico’s agency on aging informed her strategies in the chaotic early days of the crisis.