Why is Tom Hanks firing at the tank in Saving Pvt. Ryan?

It was definitely the last resort of a desperate man.
Broken and dazed, his company in tatters, Miller doggedly pulls out his pistol and fires at the tank in a futile but defiant gesture.

His mission was over, Ryan had been found and given the message, Miller's trusted sergeant lay dead at his side, as far as Miller was concerned this was the end of the road.

So, in answer to your question, it was a combination of resilience, defiance and resignation that caused Miller to unload his pistol at the approaching tank.

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