Although I think that this is ultimately more of a political question than a religious one (asking for opinion regarding the validity of a certain amendment), it might be able to be answered to a certain extent.
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside. No state shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any state deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
It should be made clear that the 'privileges' that are not to be denied to any individual but rather protected equally for all are those privileges that are naturalized within government law; this includes the freedoms and rights that are displayed elsewhere in the Constitution (freedoms and rights that shall be later referred to as primary privileges).
For Christians against same-sex marriage being manifested in any law (be it state or national), there is likely a belief that marriage is firstly the privilege of the Church, and not of the individual. In other word, the right of marriage belongs not to those being married but to those doing the marrying, and the definition of what is meant by 'marriage' likewise belongs to such an institution. It is/was the Church that marries two people, and the government bases/based its laws of recognition of such a marriage around the privilege belonging to the Church. This primary privilege involves the freedom of institutions to describe marriage as they so please; the Church understands marriage to be the union between a man and a woman through use of natural reason and divine revelation, and this same understanding was held by government for a great while. For much of Western society, marriage was primarily a theological term, and some (and only some) of its implications were captured and sanctioned by civil law. But beyond Western society we still find that the cultural understanding of the bond between two lovers was predominantly understood firstly through the lens of some institution within the natural and free market, most often times being religion. It is based on the institution/religion's consideration of marriage that civil law, if there was to be one, operates. The relationship is rarely turned the other way around historically. The result of government sanctioning 'marriage' typically is/was seen in additional privileges in the economic system thereafter to married couples. Note that this entire discussion appears to be very subjective, and it is. What is shown is not an actual argument for what definition or understanding of 'marriage' is the right one; what is being discussed is how government relates to religion regarding marriage. We find that typically government operates on an understanding of marriage as established by a preexisting free market in which religions, more likely than any other institution, establish what 'marriage' is.
All this tells us very little about our own government system in regards to marriage, but it should be noted that the government was likely to fall to this same pattern. 'Marriage', if nothing else, is a word that is used by all kinds of different people to describe different things. Our national government made little effort to establish in the beginning their own 'definition' of 'marriage' by which all people had to prescribe to. But that being said, the culture at the time was greatly influenced by Christianity, and the only 'marriage' that existed in large was that which was established by and in the Church. Government eventually got to the point in which they 'recognized' marriage, and this marriage was between a man and a woman, as described by the Church. This is not an argument that the Church is right (that requires another line of thought). It is only establishing a fact, and the nature of the relationship government has with religion. This being said, from then on the typical additional privileges that accompany government sanctioning a thing followed. The most predominant of these privileges are economic in nature. Of course, this very action of assistance would lead to certain individuals who possess different opinions about sexual union and marriage asking why they don't have similar additional privileges in the economic system (it should be noted that additional privileges are not sanctioned necessarily to every individual, nor is such the correct interpretation of the 14th amendment which regards primary privileges). So in our modern society, being the sympathetic country that we are, we have sought to resolve this discrepancy by stating that these individuals can be married too. But of course, this decision did little to consider the nature of primary privileges as regards marriage (and the idea that such a privilege belongs to those doing the marrying) and the nature of the moral argument itself (we have not even discussed yet what 'marriage' truly is). The latter is not entirely necessary for one to take problem with the SCOTUS decision. The relationship between government and the free market of religious institutions has radically altered; where marriage used to be in some sense a meaningful term as generated by the institution doing the marrying, it has now become an entirely subjective term (since the individual, or one being 'married', can decide its meaning) describing a union mostly generated by government action.
So in asking how Christians that don't support the SCOTUS decision but do support the 14th amendment interpret both cases, the answer has already been given; we find that marriage is not based primarily in government law, but rather primarily in natural law. The 14th amendment serves to ensure that all rights and freedoms sanctioned elsewhere in the Constitution are not violated by state government. But the privilege of marriage includes both primary and additional right. The primary right belongs to those doing the marrying (one consequence of such is that they generate the meaning of the word 'marriage'); the additional right is distributed by the government for the individual, an action that is always in subjection to the primary right (as established in the very structure of our own government, along with the majority of other governments). The privilege of marriage (according to natural law) is rightly that of the Church, and not of the individual, for it is the Church that performs marriage. But the privilege of 'marriage' (according to modern society) is rightly that of the individual, but such an interpretation is grossly flawed, as shown above.
This is my understanding, and I believe that if most traditional marriage supporters could verbalize more precisely their thoughts they would share this view. This answer has established mostly a political argument rather than moral argument, upon which all things rest. But I think that a political answer better fits the nature of the question.