I think the military would take sides, but it wouldn't overthrow the government.
I think you would see various states activate their National Guard and take control of the situation, and await Congressional resolution of the issue.
Some states might not, and they will suffer terribly for it. But, the rest of the country will be intact.
...but it wouldn’t overthrow the government.
If Congress does the right thing and uphold the law not a political party. If they did support party over Constitution down they would go too.