Can an employer force an employee to write a letter of apology?

No. But an employer can, in most cases, fire an employee for not writing one. The "at will" employment doctrine means an employer can fire an employee for pretty much any reason they want so long is they don't violate specific anti-discrimination laws. Since there's no anti-discrimination law that covers apology letter writing employers have free hand. The exception might be if you have an employment contract that spells out in greater detail elements of your employment such as term of employment and detailed job responsibilities. In such a case the employer could still fire you but you might have a case in court for contract violation.