Shouldn't 'I pay the meal' be accepted?
But that sounds like you're giving the money to the meal, rather than paying someone else (unspecified) for the meal.
Is this correct? There's no need for a preposition?
Yes it is. Could also be "Πληρώνω για το γεύμα" with a preposition but both are correct.
Plus, if you mean something like "it's on me, my treat," there's a whole new verb for it: Κερνάω εγώ. The stress on εγώ.
Επιτρέψτε μου αυτή τη φορά να κεράσω εγώ.
Κερνάει ο Γιάννης. Έχει τα γενέθλιά του.