It is an American expression. Americans 'think' that they speak English, but they frequently do unusual things to English grammar. It means - I am unable to meet you on Thursday. Other English speakers would say - I am unavailable on Thursday. I am not able to meet you on Thursday.

Yes. I can't do tomorrow. I can't do next Friday. I can't do lunch. Etc.

Thank you. Is it a common phrase?

Yes, that is exactly right! :)