NO: Lilienthal has the right answer for non-contract work, or at will employment (no all US states are at will employment, but the same idea basically applies. If there is no set term of employment, either side can terminate the arrangement).
YES: If the employee signed a contract specifying a length of time to be employed, AND that contact has specific terms for ending the employment early, AND the contact makes a difference between an employee resigning, and a employer firing, then yes they can turn down your resignation.
An example of something like that may be, tuition assistance in exchange for a 5 year contact, if the employee resigns and the resignation is accepted, then the employee has to pay 50% of the tuition assistance back, pro-rated for the length of employment. If the employer does not accept the resignation, then the employee must pay 100% of the tuition assistance back, pro-rated for the term of employment.
OTHER: Usually, the difference between an accepted resignation and a non-accepted one comes down to "why you left" in the HR department. If the resignation is not accepted, then it will look like you just stopped showing up to work. You will be "fired" for no call/no show. If your resignation is accepted then you left on good terms.
Aside from contract specifics, which are a real thing but vary a lot. I have seen companies reject resignations for a number of reasons.
- An attempt to work out a deal to keep the employee. I will not accept your resignation, but I will offer you a raise.
- An attempt to get the employee to go though the exit process. I will not accept your resignation till you hand in your security badge and take the exit interview.
- A contractual agreement, that the company did not feel the employee met. I will not accept your resignation, we paid for you to go to foo school to learn to make widgets, and you have yet to produce a single widget for us.
- A play for time, to let the employee cool off. I will not accept your resignation now. If your still upset about our change in health care providers, give it to me again next week.
Keep in mind that worst case scenario, outside a contract, is a bad reference. That's serious, but they can't force you to work or keep your pay or anything like that. Inside a contract, that's a totally different story. They can't force you to work, but they can take money from you, keep your pay, transfer bills or costs, or all kinds of other things.