The Collaborative International Dictionary of English v.0.48:

Rightfully \Right"ful*ly\, adv.
   According to right or justice.
   [1913 Webster]

WordNet (r) 3.0 (2006):

rightfully
    adv 1: by right; "baseball rightfully is the nation's pastime"
           [syn: rightfully, truly]

Moby Thesaurus II by Grady Ward, 1.0:

34 Moby Thesaurus words for "rightfully":
   appropriately, as is proper, as is right, by right, by rights,
   condignly, correctly, deservedly, disinterestedly, dispassionately,
   duly, equally, equitably, evenly, fair, fairly, fittingly,
   impartially, impersonally, in all conscience, in reason, inequity,
   injustice, justifiably, justifiedly, justly, meetly, properly,
   right, rightly, upon even terms, warrantably, warrantedly,
   without distinction