Let me ask this : what was the legal reason for denying marriages to immediate family members prior to Obergefell? Was it just tradition? Potential genetic abnormalities? Are those the only two compelling state interests for preventing consanguineous marriage?
Because of the potential risk of birth defect in close relation offspring. Believe it or not, this has been challenged in state law through the years. Sibling marriage is not permitted because of this potential risk... but "marriage" has now changed and is no longer defined as a male-female union, so the entire issue of procreation becomes moot.
Also, "traditional values" are no longer applicable. Your "fears it may lead to" arguments are invalidated by an established constitutional right of the individual now. Obergefell did not happen in a vacuum.
It seems that if birth defects were the only reason, infertile consanguineous couples might have already been given leave to marry.

If cases have been ruled on in the past and defects are the only reason ever given, the state wouldn't have a precedent to use to prevent same sex consanguineous couples, at least.
Obergefell did not establish marriage as a constitutional right. That was done through multiple previous USSC rulings.
That something is not considered a strong enough argument in one instance does not mean the same will be true in others. The potential harm from consanguineous marriages, particularly regarding parents and children, will almost certainly be seen as stronger than any potential harm from same sex marriages, especially considering the reason for the potential harm is different.
Well. if you go research this, you will find that laws vary from state to state and country to country. There is currently no state that allows sibling marriage. Each state has it's own statutes to address incest relationships of which "marriage" is defined as a prohibited act associated with the behavior. However, it is difficult to ascertain the specific "justifications" for these statutes, they simply state that it is illegal. And as I said, they do this in different ways... for instance, in North Carolina, the issue of incest, and subsequently, sibling marriage, is covered in the Public Indecency Act along with a host of other behaviors such as adultery and "lewdness", some of which have already become legally acceptable under other laws and are no longer subject to the act. We're dealing with a lot of antiquated and outdated legislation passed sometimes over 100 years ago in a different era. I think the act from NC was originally passed in 1868.
Now, I have always assumed, and I am pretty sure most people have as well, that the main reasoning and justification for bans on sibling marriage is the concerns associated with birth defects of offspring. You want to shoot that down and demand that I present something else and I don't honestly know of any other reason... do you? Aside from the fact that it's frowned upon as taboo in society... but the same could be said for homosexuality just a short time ago. In fact, I believe homosexual acts are covered in the aforementioned North Carolina Public Indecency Act.
That something is not considered a strong enough argument in one instance does not mean the same will be true in others.
That's exactly what it means when the argument is the same. The 14th doesn't say it applies only when you want it to apply and not whenever you don't.
The potential harm from consanguineous marriages, particularly regarding parents and children, will almost certainly be seen as stronger than any potential harm from same sex marriages, especially considering the reason for the potential harm is different.
I've not seen you or anyone else explain what this "potential harm" is... other than some moralistic viewpoint which is fundamentally no different than the same moral arguments against gay marriage. You need to come up with something substantial and compelling here and you simply haven't. You keep talking in grandiose platitudes about morality and decency like a Baptist evangelist but you're really presenting a piss poor legal argument that will withstand a court challenge in 2015.
When it comes to parents and children you have a person in a position of authority over the other. That creates a greater danger of abuse and exploitation than other relationships. I have compared this to teacher/student relationships in the past. Teachers are not allowed to have romantic relations with students even if they are of legal age. While not necessarily a criminal act, I think some of the reasoning behind preventing such relationships and preventing parent/child romantic relationships is the same; again, one person in a position of power over another.
I've said before it becomes more of a muddied argument with adult siblings. There can still be that danger of one sibling having been an authority figure over the other, but does that rise to a level of danger strong enough to constitute a compelling state interest? I don't know.
There is also the fact that marriage involves the creation of a new immediate family unit, legally speaking. If the participants are already immediate family members, the state might argue that creating a new family unit is part of the requirements for marriage and that is reason to prevent consanguineous marriage. That might well depend on how various marriage laws are written or perhaps previous court rulings setting some sort of precedent.
I agree that a good deal of the reason for preventing such marriages is fear of possible genetic issues and social taboos. There are other possible reasons as well. Whether any of them would stand up in court is not something anyone can say for sure. One big difference is that I don't believe there are very many consanguineous relationships existing and so the likelihood of cases being brought to court are less; along with that, while homosexuality in general and same sex marriage in particular have seen a gradual increase in acceptance over the past decades, there has been no such growth in acceptance of consanguineous relationships that I'm aware of. Part of what pushed same sex marriage through the courts, and potentially influenced the decision, is that society was ready to accept the Obergefell decision.
What 'grandiose platitudes' have I used? Talking like a Baptist evangelist?
What potential harm from same sex marriage was argued prior to Obergefell? How were those arguments 'fundamentally no different' from those presented regarding consanguineous marriage? If your answer is that they are based on moral judgements, as I've said before, all law can be broken down to moral judgements. Unless you want to say that all law should be invalidated, I fail to see the point.