Can the Remove Sickness spell suppress the sickness effect that results from an Oracle with the Legalistic cure breaking his word? Or does the constant nature of the curse trump the spell's usual suppression?
Remove Sickness would suppress the sickness for it's duration (10 minutes/level).
Legalistic doesn't state anywhere the sickness is impervious to magical effects.
As far as I can tell it's magical sickness with a variable duration (24 hours or until you meet your obligation.)
Per the rules, Heal or something similar to it would straight up remove the sickness.
As a DM though, I would feel a bit irked at a legalistic oracle constantly shirking the penalties of their curse. Perhaps whatever power caused the oracle's curse in the first place would make a visit.