Choose the correct form---

The correct form is:
He fell down while he was walking.
Explanation:
The Grammar Rule: When one action (the interruption) happens in the middle of another longer action (the background activity), we use the Past Simple for the short action and the Past Continuous for the long action.
Short action (Past Simple): He fell down.
Longer action (Past Continuous): ...while he was walking.
Why the others are incorrect:
He fell down while he walked: This is sometimes used in informal speech, but it is less precise. "While" specifically implies duration, which the continuous form (-ing) expresses best.
He fell down while he was walked: This is passive voice (as if someone else was "walking" him). It doesn't make sense here.
He fell down while he walking: This is grammatically incorrect because it is missing the auxiliary verb "was."