Mr. Abbott has said that with so many counties unaffected so far by the coronavirus, what works in urban Texas does not necessarily work in rural Texas. He said on Tuesday that his primary obligation was to public health and safety, but added, “If the goal is to get the economy going, the best thing we can do to get the economy going is to get Covid-19 behind us.”

Mr. Abbott has taken a series of steps to curb the spread of the virus, including deploying the National Guard, banning dine-in service at restaurants, shutting schools and limiting social gatherings to groups of 10 or fewer people. He suggested that more severe measures may be on the way, remarking that he was surprised to see how many drivers were on the road when he was on his way to the news conference in Austin.

“It is clear to me that we may not be achieving the level of compliance that is needed,” he said, adding, “We will continue to evaluate, based upon all the data, whether or not there needs to be a heightened standard and stricter enforcement.”

Mr. Abbott’s remarks on Tuesday came amid a national backlash over Mr. Patrick’s statements on Fox News, in which he echoed comments by the president, who had expressed hope that the economy could be restarted sooner rather than later.

Mr. Patrick took it further.

“I’m not living in fear of Covid-19. What I’m living in fear of is what’s happening to this country,” he said. “And you know Tucker, no one reached out to me and said, ‘As a senior citizen, are you willing to take a chance on your survival, in exchange for keeping the America that all America loves for your children and grandchildren?’ And if that’s the exchange, I’m all in.”

On social media, humorous GIFs about whether the old should sacrifice themselves for the young spread like wildfire, and #NotDying4WallStreet began trending on Twitter.

In Texas, some elected officials were silent on the lieutenant governor’s stance, while others scratched their heads.