Although this is illustrative of bad grammar, it is somewhat logical.

Problem:
The contraction “don’t” is not a contraction of “does not”.

Explanation:
Forget about the source of contractions while you consider these plural forms:

• We won’t. We don’t.
• You won’t. You don’t.
• They won’t. They don’t.

Now, consider these singular forms:

• I won’t. I don’t.
• You won’t. You don’t.
• He won’t. He doesn’t.

In other words, only the third-person, singular contraction for not doing something uses “doesn’t” instead of “don’t”.

The reason for this is that the third-person, singular, non-contracted form for not doing something is “does not”, whereas all of the other non-contracted forms are “do not”.

From an outsider’s perspective, though, “He don’t.” is somewhat logical, especially when going by sound alone and when failing to remember that “don’t” is a contraction of “do” and “not”.

For fun, I searched Google for each of the following phrases (with the quotation marks) and got about the indicated numbers of matches:

• “He doesn’t” — 5,770,000 matches
• “He don’t” — 5,230,000 matches

This tells me that Web authors have written the phrase correctly vs. incorrectly by a ratio of 1.10:1, which is a bit pathetic.

Solution:
“He doesn’t.”