If a product cuts a consumers finger and requires stitches, is it the stores fault or does the manufacturer have to pay for the medical bill?

Generally, there is a lot more to the answer than meets the eye.  Your question falls under the category of products liability.  Under the typical laws in each state, one has to decide if this is a dangerous product, if the underlying product was designed with a defect, did it leave the manufacturer with a defect, did the retailer somehow alter the product, etc.

What caused the cut? Did the consumer use the product negligently? Did the product include directions for use or installation that the consumer did not properly follow or ignore?
Without more information about the product itself and the circumstances surrounding the cut and stitches, the consumer may or may not have a claim against the store or manufacturer or designer or distributor. It follows the stream of commerce, but if the product was altered along the way, the act of altering has to be reviewed to determine if liability exists therein.

The manufacturer and the store are both liable for your injury caused by the defective product.  The manufacturer, seller, distributor and anyone else who had a role pertaining to this product would all be liable for your injury.  This area of law is called products liability.

You should document your injury by obtaining the medical records and medical bills.  Your claim should be filed separately with the insurance carriers for the manufacturer, seller, distributor, etc.

In addition to compensation for your medical bills and wage loss if applicable, you should request additional compensation for pain and suffering.