Seagrave's argument is that marriage is not an act of one person only, but necessarily involves the participation of one other person, and that rights of the kind guaranteed by the Fourteenth Amendment pertain only to isolated individuals. The first part is of course correct, and the second may or may not be correct, but his conclusion is both irrelevant and misleading.
For one thing, the language of rights in the Constitution in general is not confined to the actions of individual persons. Take, for example, the First Amendment, which refers to "the right of the people peaceably to assemble". An isolated individual cannot "assemble"; only a group of persons can.
Secondly, of course an advocate of "gay marriage" (which is what this is all about) could even accept Seagrave's assertions and still say, "But no one has a right to stop two men from marrying each other." After all, early in his essay Seagrave mentions that he would like to be a millionaire, but that he has no right to be a millionaire. I think he would agree that although the state may have a right to prevent him from becoming a millionaire by certain means (like theft or blackmail), the state probably has no right to simply prohibit him from being a millionaire.
Thirdly, Seagrave's comments really only apply to mutable positive law. Until 1971, there was no "fundamental right" of an eighteen year old American citizen to vote, but with the passage of the Twenty-Sixth Amendment there now is. (It is not, of course, an absolute right -- a convicted felon typically may not vote, for example -- but it is "fundamental" in Seagrave's sense.)
It is really a mistake to talk about "gay marriage" as though it were about rights at all. The question is not, "Should we grant to every couple the right to marry, regardless of who the two people are?" or even "Does every couple have an inherent and inviolable right to marry?" The question is about "can every couple marry", not "may every couple marry".
For an analogous situation, suppose for the sake of argument we take the word "couple" to mean "two people in a romantic relationship". Someone would inevitably find that too restrictive; suppose that the preferred solution were to define the word "two" to mean "any number greater than one". It could of course be argued that there is no inherent meaning to either the sound or to the sequence of letters (or even the letters themselves); these are culturally defined. It could be noted that some cultures have words for only a handful of numbers, and that English itself distinguishes singular nouns from plural nouns, using the plural for any number greater than one. It could be noted that we have often changed the definitions of words; for example, "fish" used to simply mean, "an animal that lives in water" -- hence jellyfish, starfish, crayfish, and shellfish. What could possibly be wrong with choosing, as a culture, to adapt the language to be more inclusive? After all, the old meaning of "two" is subsumed within the new meaning of "two", so how could this possibly hurt existing couples?
Video added 7-5-2015.
The obvious answer is that we would lose arithmetic. If we redefine "two" -- if we try to not merely change the word but the whole concept, as is being attempted with marriage -- we lose arithmetic. Under the new rules, 2+2=2 and 2-2 is undefined. We would lose our ability to reason meaningfully about numbers.
From the point of view of tradition, redefining marriage would do for anthropology -- the study of man -- what redefining "two" would do for arithmetic. Our ability to understand human beings properly would be impaired, and our ability to make rational decisions relating to human beings would be compromised.
Finally, Seagrave unintentionally invites questions such as, "If there is no fundamental right to marry, could the state abolish marriage altogether?", "Would it be fundamentally wrong for the state to forbid any Christian from marrying?", etc. Even granting the right to a couple, perhaps there are grave circumstances which might prohibit a valid marriage, as there are grave circumstances which might prohibit an adult citizen of the United States from voting, but these would have to be grave reasons. The problem with "gay marriage" is that it is not really marriage, just as 3 is not really 2; the problem is not that marriage is some sort of privilege which the state is free to grant or withhold at its whim.