WordNet (r) 3.0 (2006):

moral obligation
    n 1: an obligation arising out of considerations of right and
         wrong; "he did it out of a feeling of moral obligation"

Bouvier's Law Dictionary, Revised 6th Ed (1856):

MORAL OBLIGATION. A duty which one owes, and which he ought to perform, but
which he is not legally bound to fulfill.
     2. These obligations are of two kinds 1st. Those founded on a natural
right; as, the obligation to be charitable, which can never be enforced by
law. 2d. Those which are supported by a good or valuable antecedent
consideration; as, where a man owes a debt barred by the act of limitations,
this cannot be recovered by law, though it subsists in morality and
conscience; but if the debtor promise to pay it, the moral obligation is a
sufficient consideration for the promise, and the creditor may maintain an
action of assumpsit, to recover the money. 1 Bouv. Inst. n. 623.