They should have to right to fire him, but first they should ask him:

  - Did you do it?
  - Do you have any evidence that you are innocent?
  - What is the evidence against you?

They should think about whether they believe he did it. In some cases that will mean firing people who are only accused and that won't be immoral.

In this case, if they just talked to him for a bit and gave him a few days to investigate and gather evidence, they could have avoided firing him.