If a private business has a right to limit offensive speech on a social media platform in the name of moral righteousness, they have just as much right to deny service to people they find objectionable on the same grounds such as homosexuals or muslims.
Not if they want to do business in the the United States they don't. Religious groups are a federally protected class, and with very few exceptions, cannot be denied service on that basis. Sexual orientation is gaining traction as a protected class at the state level; any web-based service discriminating on that basis is likely to run afoul of some of these states' laws (for clarity, "public accommodation" in the context of that map means a business that is open to the public).
Would you be OK with ISP's being pressured by moral crusaders to not provide connectivity to people who host "offensive" content because the moral crusaders decide to label everyone they don't agree with "neo-nazis"?
No, I wouldn't be OK with that because the moral groups would have no standing or injury. Unlike YouTube and its advertisers, the moral groups in your scenario are not party to any contract with the ISP or its customers. Further, I'm of the opinion that ISPs should be regulated as utilities and required to serve anyone who's capable of paying their bill, just like the electric company. That, I suppose, is another discussion entirely.