I stand corrected on the incorporation thing. I failed to complete my thoughts on the issue.
Obergefell v. Hodges relied on the 14th ammendments "due process clause" (immunities and equal protection) which was a bad idea for marriage rights. Because there is no enumerated right to marriage in the BoR like there is with the 2nd Amendment. This ruling has left to many unanswered questions (1st Amendment religous rights for one). If the SCOTUS had found that marriage was a 9th Amendment right, it would have made marrage a PERSONAL right retained by the people instead of just a desire by 5 unelected, unaccountable government officials to act like legislators.
By using "Due process clause"in the 14th A, then the right to keep and bear arms would also fall under the due process clause in the right to State issued firearm licenses with every State. Now if a State "Justice of the Peace" has discretionary power to issue marriage license (as it is in many cities/ states with "May Issue" firearms licenses) as they see fit, then the Obergefell ruling can be a very limited right or no right at all. If not, then It makes it easier to say that the rights are not being given equal protection, since every State must now recognize every other States Drivers license (by choice) and Marriage License (by SCOTUS ruling) and since every state has some form of firearms licensing to carry a firearms, then under equal protection, someone that holds a firearms license in one State would have to be recognized by the other States as well. Lest States be allowed to not recognise other states Marriage licenses as well.
The question still remains;
"'on what grounds can states refuse to honor another class of license that permits a different protected right?"
If States cannot apply different rules to the (now) protected right of Marrage then the same goes for the protected right to Weapons/firearms/Arms. However in the case of marriage, they used the 14th Amendments 'due process clause" instead of the 9th A, (or in the case of firearms the 2nd A) then the current protection given to marriage leaves the States with the ability to heavy regulate that Marriage License. Due process changes over time. For example, I see no reason why it would be illegal for a State to make additional requirements to get a marriage license such as additional training, testing or other non-dicriminatory requirements that could/would lead to a refusal to grant a marriage license to many people both gay or straight. An example would be; mandating genetic compatablity testing to protect the public from having to support the potential handicapped offspring of genetic disorders like Cystic Fibosis, Down Syndrome, Progeria and soon on. This testing would not evaluate whether the couple can conceive, but rather, if they are compatable to produce healthy offspring as a couple. It would apply to everyone that is appling for a marriage license thus it would also serve to enforce the laws on incest which is already illegal in every State. However, such testing would also deny Gay marriage much like "good cause" does for firearms.
The second issue with Obergefell v. Hodges is that Marriage cannot be an Indivual Right. I can speak, pray, petition the government for redress, keep and bears arms, remain silent, be secure in my home etc. without any assistance from anyone. When people say they have a right to marry or right to healthcare or food stamps etc., its not true because if no one agrees to marry you then you can't get married to yourself, if the state refuses to issue a marriage license then what? The authority to grant something also includes the authority to deny it. If all the doctors and nurses leave your state, then your right to access and receive medical care does not exist.