The answer is probably 'yes'.
A contract doesn't need to be in writing for it to be legally binding. If I call in a builder and we verbally agree that he'll do a certain job, for which I'll pay a certain price, we've both entered into a legally binding contract. If we were to put that contract into writing, the absence of a date wouldn't nullify the validity of the contractual agreement which we'd already entered into verbally.
If the person who failed to add a date to his signature wanted to get out of the contract, he could try to convince a court (if the matter got that far) that he'd deliberately omitted the date with the full knowledge of the other party because he'd only given his signature as an indication of good faith that he'd fully enter into the contract (by dating the document) at a later date, if certain conditions were met by the other party. However, unless he had supporting evidence, it's likely that a court would disregard his argument and declare that the contract was still valid.
Chris