“It’s all Tom’s fault.”
The ability to admit that you are wrong, without justifying and blaming others, is considered to be a sign of a mature person. This employee evokes trust and respect, even if they sometimes make mistakes. And if it was really Tom’s fault, there are dozens of other options to unobtrusively mention this, without it becoming personal, and only when the outcome of the situation depends on it. It’s a well-known fact that telling on someone doesn’t look good.
A better way to say it: “You’re right, I missed that detail. From now on, I’ll be more careful. Maybe we can correct it this way…”
If you decide to leave, just leave. If you keep using slamming doors as a tool to uphold your own opinion, after a couple of these tantrums people will stop taking you seriously. It’s known that there is no such thing as an irreplaceable person. So don’t be surprised if, after closing the door this way, you soon find that nobody will let you in again.
A better way to say it: “It seems we have different opinions on that.”