The FCC does have the authority to regulate, but they did it wrong technically. If ISPs are classifieds as common carriers, then the FCC does have authority. ISPs are not currently classified as common carriers, so the FCC does not have authority to regulate them as such. The FCC can re-classify them as common carriers, and then would have the authority to make such rules.
The ISPs used to be classified as common carriers, but the petitioned the FCC to reclassify them in the mid 2000s as non-common carriers. The ISP's arguements were that they provided services, not just internet access. They provided: email, web hosting, etc. In my opinion those arguments were not valid. They did provide such services included with internet access, but I did not know one single person who chose their ISP based upon email or web hosting. I know very few people who actually used the ISP email or web hosting. Now, I don't believe many ISPs offer web hosting included in the price of internet access. AT&T has stopped providing email services and sends its users to Yahoo mail.