Ironically, I think the constitutional issue here is the simpler one.
Remember that from the Christian world view, marriage is established by God between Adam and Eve. It is a religious structure before it is a social or legal structure. Therefore, by the reasoning of the poster in the question, any law or regulation that recognizes marriage — at all — is unconstitutional. That would include not just who can marry whom, but also joint filing for taxes, federal employment benefits for spouses, federal tax breaks for private employment benefits for spouses, immigration rules, health privacy rules, joint property laws, and much more.
On the other hand, the secular (a-religious) view is that God could not have established marriage (because this view doesn't allow for the existence of God in the first place). Therefore marriage must have first arisen as a social or legal structure, which religion later co-opts. If marriage is primarily a social or legal structure, then society or government should be allowed to set it's bounds at will. Any other restriction would be the same as legislating religious compliance, which, while often tempting, is probably not a good idea.
Fortunately, there is a third category. The poster in the question is fundamentally flawed, because it focuses in on the word "religion" and ignores the entirety of the phrase "establishment of religion". This completely changes the constitutional implications. We must now also think about what constitutes an "Establishment of religion".
On the surface, marriage might qualify as an "establishment of religion". In that case, the poster is entirely invalid; not only is Congress prevented from making any laws about marriage, but as a confirmed religious establishment the definition is left entirely up to religion, and government has no say at all. If religions choose to exclude homosexual marriage, then that's what marriage is and Congress has no power to do anything about it. It's also worth noting that all tax, inheritance, immigration, employment benefits, etc outlined in my first paragraph would be impossible... at least at the federal level. Part of the premise of the poster is to first accept the word "sanctity", and the result is that this is the position the poster supports most strongly... more so than legalized homosexual marriage.
A better legal interpretation here is that marriage is not an establishment of religion, at least in the Constitutional sense. Instead, the Constitution is referring to different religious groups: the Catholic church is an establishment of religion, the Southern Baptists are (maybe have?) an establishment of religion, the Mormons are/have an establishment of religion, and so on, such that even Islam, Hinduism, and Buddhism are establishments of religion and equally protected. The purpose, then, of this clause is to prevent the state from favoring (or disfavoring) any one religious group over others. The founders did not want another Church of England telling them how to worship. This clause ultimately has nothing to do with marriage at all.
Put another way, the creator of the poster likely sees religion trying to use government to create new restrictions on marriage, and believes that the constitution prohibits this. On the other hand, many Christians see the lbgt community trying to use government to expand marriage to mean something it has never meant before, and use the exact same reasoning to argue against it. Christians could easily create a poster with most of the same text that argues the exact opposite (and by acknowledging the whole "establishment" phrase rather than just the one word, would probably have a better interpretation of the constitution in the process). However, I think I've shown here that both views are flawed.
The last interpretation does allow some room for Christians to argue that marriage is primarily a religious matter, that the precise definition is not something that society is free to expand at will, and any laws defining or using marriage relationships are merely acknowledging what exists elsewhere. However, I think the current legal landscape is against Christians here. Marriage may be a religious structure, but the establishment clause does not create any protected legal status for marriage, because marriage is not shown to be an "establishment" of religion. The state has the power to recognize any marriage it wants to, even if it is clearly counter to the traditional definition. Constitutional issues aside, Christians are free to argue that the man-woman definition of marriage is the status quo and that expanding that definition is a bad idea (and I believe they would be right to do so). Unfortunately society and government no longer recognize God's authority on this matter.
The good news is that I believe that the state does not have the power to force establishments of religion to recognize these so-called marriages. Therefore, perhaps a better response by Christians would be for our establishments to more strongly assert that right.
We can now also answer the religion vs worship question, using some of what we learned addressing the constitutional issue. Christian religion would be an "establishment". Worship would be the practices of the individuals in the establishment.
