It sounds okay. I would add the before the word pain because you're specific that the pain is from tooth extraction. You can also say Sorry, I had my tooth removed. I can't go out anywhere because of the pain.

I'd say it this way: "Sorry, I had a tooth pulled and it hurts too much for me to go out."

My tooth *has been* removed. I would use the present perfect tense here, because it happened in the past and is true today, which is your reason for not going out. But it would be better and more natural-sounding to say: ‘I had a tooth removed’ or ‘I’ve had a tooth removed’ this is how a native speaker might say it). Because of *the* pain: A definite article is needed here because it is the one specific pain you have. A definite article (the) is used before a noun to indicate that the identity of the noun is known to the reader.