Generally if you threaten to quit you should be prepared to quit. In the US most employment is considered "at will" unless a contract specifies otherwise (and a few states have different laws). Meaning employer or employee can terminate the agreement at any time for any reason or no reason. The exception to that is if you were fired for prejudicial reasons and you are a protected class (sexism, racism, etc). That means I can quit today with no notice required. It also means my employer can hand me my last paycheck today and tell me to take a hike.

An example of this. My wife's employer recently found out she had applied for a job at another company. They fired her that day. She didn't get the other position and the previous employer is blocking unemployment compensation.