Without knowing more facts, it does appear that this employee "quit" versus was "fired." So be careful not to use the word "fired" unless you want to "muddy the waters" with the state.
Employees who are fired for reasons other than "cause" are normally able to file for and collect unemployment benefits. "Cause" normally is especially egregious behavior such as dishonesty (e.g., stealing or having a history of falsifying time records), behaving in a violent or threatening fashion, etc. In other words not simply poor performance.
Often, employees who are fired for valid reasons such as constantly being late for work or being insubordinate file for unemploymen. And even though the employer well responds to the state's inquiry and justifies firing the employee, the state may still determine in their infinite wisdom that the employee is entitled to unemployment benefits.
Employers often appeal those decisions, and sometimes they win--but it can be time-consuming and frustrating.
You have to decide what risk you are willing to take in this situation. Do you want to "accept the employee's resignation" (submitted via leaving his workstation and/or the premises without notice or explanation) and risk paying for his unemployment benefits or do you give him another chance?
If it is the former, document the incident and be prepared for the state to ask for your rendition of what occurred. Remember not to use the word "fired" when you do that. Good luck.