It is good practice for an employer to confirm a verbal warning, even though that sounds daft.
Since the employer is using the first stage of the disciplinary procedure, yes, you should be allowed to appeal it. What does the disciplinery procedure / grievance procedure for your company say about it? Your company doesn't appear to have one? - I'm not surprised.
Personally I would let it lie, however, I suggest that you WRITE YOUR OWN NOTES of what happened, sign it date it, and keep it safe.
If things get nasty again later, and there is reference made to the earlier warning, you have your notes to refer to - and the employer probably will have nothing.
Verbal warnings typically last 12 months - again the Code should say - again if you weren't told, that is poor.
Just be aware that an awful lot of cases that come to Employment Tribunal fail for the employer because they did not a follow a reasonable or fair PROCESS, then fail because the ACTION by the employee was inappropriate.