Employment Law Questions? Ask an Employment Lawyer.
Yes, technically they can terminate you. Just because the supervisor accepted your apology, that does not legally mean that they can't use the incident as a basis for termination. Furthermore, the employer as a whole can decide that they believe the event is basis for termination even if the supervisor does not. This is because the employer is not bound by the decision of the supervisor, in accepting your apology.
Unless you had some evidence that the employer is treating you more harshly than others (who have done the same thing) based on your race, religion, gender, age, disability or recent FMLA use, I do not see any other legal options here.
There is not a time limit, like a statute of limitations. They are not a government and this isn't a criminal law action.
In terms of a right to ask HR for an investigation, nothing in external law creates that right..no. For you to have any sort of right to an HR investigation, that has to come from company policy. Only a company policy or handbook could legally create any right to an HR investigation, so you will have to explore your own company policy on that issue.