Wrongful termination occurs when an employer fires their employee for a reason prohibited by the law. This includes termination based on discrimination, retaliation against whistleblowing, violation of public policy and breach of contract.
First, you might have to file an administrative complaint with a government agency first. If the employer's reason for termination were unlawful, then the employee can bring a wrongful termination claim. This is even if the employee is an at-will employee.
"Pink slip" or work separation notice - optional in most states - not required in Texas (however, giving at least a simple work separation notice can help prevent ex-employees from filing wage claims based upon "work" they allegedly did after your company thought they were gone - see comment 5 under "Work Separations - ...
Can you get fired without a written warning in Texas? Yes, in Texas, employers have the discretion to terminate employees without a written warning.
Texas, like many U.S. states, operates under the “at-will” employment doctrine. This means that unless a contract or an agreement states otherwise, both employers and employees can terminate the employment relationship at any time, for any reason, or even for no reason at all, provided it's not an unlawful one.