Can I sue the bank for refusing to cash a check drawn on its own bank if I do not have an account with them?
No, because you don't have a contract with the bank. You can complain to the party that issued the check to you, if you wish. That party could have a case against its bank for its refusal to honor the check, depending on the circumstances.
The bank may have had a problem with your identification, in which case it's actually looking out for its customer's interests, as well as its own. There may also be limited cash available at the branch you approached, and the bank therefore was unable to cash a check as large as the one you presented. There might be other valid reasons for the bank's actions, or lack of action.
BankingQuestions.com is a free service made possible by the generous support of our advertisers. Advertisers are not responsible for site content. Please help us keep BankingQuestions.com FREE by supporting our advertisers. When you see an ad for a product or service you may have an interest in, click through to learn more.