No, the airline will not transport a non-Visa Waiver Program eligible traveler to the United States without a visa or other appropriate travel document (doing so is a violation of law and will result in a fine for the carrier). If you do not have a C-1 (or B-1/B-2 or potentially other visas, which can also be used for transit), you will not be allowed to board the plane in the first place.
You should contact the airline immediately and ask for an alternative routing. If the airline can't do this, you will need to plan this yourself and you need to get serious about checking visa requirements for potential transit in other countries. This WikiVoyage article may be helpful in avoiding US connections.