In keeping his self-made promise to go fast and not stop, the hare took off and ran at top speed until he came to a broad river.
That is a perfectly well formed sentence in English.
A promise to do something is very idiomatic. "To" is used to show a purpose or goal.
Question"Is "promise going fast and not stopping" grammatically correct?
No, it is incorrect. Correct is: The hare promised himself to go fast and not stop.