The general form of the argument is:
If P, then Q.
P is a fallacious argument.
Therefore, Q is false.
The problem with this line of thinking is that even though an argument is fallacious it may still have a conclusion that happens to be true. In other words, a bad argument doesn't make a conclusion automatically false. It just means that the argument presented does not provide a good reason to believe that the conclusion is true. It's still entirely possible that the conclusion is true but that it is true for reasons not given in the argument.
Though Tom's argument is fallacious this is not proof that he is not English.
Wikipedia: Argument from fallacy
Fallacy Files: Fallacy Fallacy
Fallacy Files: Bad Reasons Fallacy