No! We are sorry to inform you that we feel that your contractor has duped you. According to the Uniform Building Code, composition shingles
MUST be applied over
UNDERLAYMENT (felt paper) except, and we quote, "underlayment may be omitted [when applying a new roof] over
EXISTING roofs except where the roof pitch is less that 4 inches in 12 inches [4" of roof rise per each 12" of horizontal roof travel]."
Solution: Your contractor should remove the roof he installed, apply proper underlayment, and finally apply new shingles. Although he may be able to save some of the shingles he removed, we strongly advise that you insist on all new shingles as a replacement since that is what you originally paid for.
If your contractor continues to be uncooperative, we recommend that you file a formal complaint with the Contractor's State License Board. If he is unlicensed, you might also contact your local district attorney. In case it, seems to us that you have cause for civil suit for the value of the solution we mentioned.