No, the public acceptance will not change. Being told that you cannot go 200 miles to see your kids/grandkids will be acceptable? Being told that news organizations cannot travel to collect and disseminate news will be acceptable? If bloggers are allowed to travel then everyone with a Twitter or TikTok account will claim to be a journalist. If only "accredited" organizations are allowed to travel, who decides who is accredited? Is MSNBC? If Fox? I Michael Moore? Is Alex Jones?
If people were asked to stay home for some amount of time, many probably would. If infected people or suspected to be infected people are forced to quarantine, most people will accept that. If the government tries to use force to enforce a mass quarantine, or general isolation, it would face a lot of opposition from the public. Libertarians would object for civil liberties reasons. Republicans would believe that the government is being too intrusive. (actually the same as civil liberties, but they wouldn't see it that way.) Democrats would oppose civil liberties intrusions, and would also suspect that the President was going to use force to disrupt the November elections.
Many churches will suspend services if they are asked to for health reasons. Some won't. Many businesses will take actions to limit person to person contact, as many have already. Many people will stay mostly at home if they are asked to and given good reason to. Some won't. If it went as far as martial law, then when government demands, and threats of force become involved, people will not tolerate it. Maybe for a very short period, but not for long. Also, the police are local, and the national guard is made of people from the community. If a national guard member was told to shoot a person from his hometown because that person is out trying to gather food for his kids, he won't do it. The government would lose control of enforcement mechanisms. There are measures between allowing people to do anything and using weapons to keep them inside their homes. However, compared to many other countries people across the political spectrum have very little tolerance for the government telling them what they
must do.
EDIT: Here is an article that does a pretty good job of discussing mass quarantines in the US. I haven't encountered the author before, but I think the article is written pretty well.
https://www.healthaffairs.org/do/10.1377/hblog20200310.824973/full/