Some thoughts in no particular order:
1) try some stripper placed on a brass bristle brush. The brass bristles won't leave scratches in the wood, but can remove some/all of the paint left in the pores.
2) on a small inconspicuous spot, apply some diluted (1 to 1) Sealcoat or shellac. When dry, apply some stain to the test area. If the stain covers the white paint in pores - Home Run! If not, remove with rag dipped in whatever the solvent is for the stain, before the stain dries. You can remove the shellac with ammonia or den alcohol if you need/want to. PS: Orange (amber) shellac adds a real nice, warm tone to walnut.
3) try #2 above with a gel stain - they act like a glaze, and are thicker.
Bob "Boardman" Borders