As I mentioned earlier today, a manifold like $S^2 \times S^2$ with a standard Morse function (giving one 0-cell, two 2-cells and one 4-cell) can never be regular for silly reasons -- the attaching map from the 4-cell is a map $S^3 \to S^2 \vee S^2$ which can't be an embedding for dimension reasons.

This example shows there's no hope for anything generic satisfying your embedding condition. A minimal necessary condition would be for the k-skeleton to be locally k-dimensional, i.e. every point in in a closed k-cell. I'm not certain if that is a strong enough condition, but its a start.

As I mentioned earlier today, a manifold like $S^2 \times S^2$ with a standard Morse function (giving one 0-cell, two 2-cells and one 4-cell) can never be regular for silly reasons -- the attaching map from the 4-cell is a map $S^3 \to S^2 \vee S^2$ which can't be an embedding for dimension reasons.

This example shows there's no hope for anything generic satisfying your embedding condition. A minimal necessary condition would be for the k-skeleton to be locally k-dimensional, i.e. every point is in a closed k-cell. I'm not certain if that is a strong enough condition, but its a start.