In management, as well as in life, you do not want to spend too much time saying what you should have done. The fact is that the incident has passed, and you did what you thought was right at the time. The best you can do now is make this a learning experience by figuring out what you will do if this occurs again, whether by this employee or any other employee. As a future strategy, when an employee engages in this type of insubordination, it makes a good deal of sense to write-up the incident immediately. If the employee wants to deny it, that is his or her privilege, and that can be placed in the file, too.
The real issue is that you have an employee who is showing signs of dissatisfaction on the job. Tardiness is one classical hallmark of job dissatisfaction, and muttering under one's breath is not exactly a symbol of a happy employee either. Hopefully you provided this employee with a clear idea of what will happen if his tardiness continues, and he should be advised of the consequences associated with any further outbursts, whether muttered, uttered, or shouted.
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