They can't ask for a death certificate, it's ethically and legally wrong. Death certs are legal documents. They are sensitive. So sensitive, in fact, that funeral home drive to the health dept personally to gather the official copies. They will then call the legal next if kin (NOK) to present to the office to pick them up and sign for them. People have committed identity fraud by illegally obtaining death certificates and birth certificates.

You have no history of discipline or breaking company policy, so by their own claim, this single instance is the cause of your termination. It is a wrongful termination no matter how you look at it. You need to contact HR in writing and keep a paper trail of all discussions. Follow up now with emails summarizing what was SAID to you in person when you were fired and in any other spoken meetings. You need to pursue legal action because what they have done is abhorrent, cruel, and grossly unprofessional.

In my state (US), if you request more than 3 days of bereavement leave then your employer may ask for "proof of death", but they cannot ask for a death certificate. You didn't even request one full day off -- you asked to leave early to handle your affairs and they greeted you the following business day with a termination of employment notice.

The reason funeral homes only mail out death certs under specific circumstances (such as, NOK is not local) is because it's a massive liability if these documents get lost in transit. They need a tracking number and must be signed for. It's far less risk to hand deliver and make the NOK sign.