Nope. The Church don't 'own' marriage and never have done. I refer you to a reply I made on asimilar topic earlier this year:
As many have said marriage is not a wholly religious institution, nor has it ever been. The concept of marriage has existed longer than any currently surviving religion and the Christian church only really took an interest circa the 13th century.
Even today marriage is inherently civil; if it were purely a magical, mystical religious ceremony then you could say whatever words you wanted, hop up and down on one leg five times, spin around, whatever your religion decreed was necessary for you to be wed and it would happen, but that is not the case. It is only when you sign the register are you married; you may have exchanged rings and said 'I do!' and everyone in attendance might believe you are married, but, legally, only when pen is put to paper and everyone that is required to sign has done so has a marriage contract been entered.
Because that's what marriage is - a contract. It's two consenting adults informing the government that they're tying the knot, so give us tax breaks, please. Religion doesn't own marriage and it never has. Sure, it performs them, but entirely on licence from the State.
If anyone 'owns' marriage, it is the State. The State shouldn't discriminate based on sexual orientation, therefore there should be equal marriage.
I used to be of the opinion that Civil Partnership was sufficient, but really it's not. Almost all the same rights are given, but it's still that little bit different: Oh yes, you are equal, you just can't get
married. It's discrimination, and shouldn't be tolerated.
I'm bisexual (the horror) so if I find a nice lady to settle down with then I can marry her, but if I find the man of my dreams and want to do the same then I can only be civilly partnered. Why?
Another issue not often raised is the fact that married couples where one partner legally changes gender have to dissolve their marriage and then form a civil partnership. With equal marriage laws this would be unecessary.
Rightly or wrongly, marriage is held up as a 'gold standard' of relationship in society and various protections are afforded to married couples in multiple international treaties, including the UN Charter of Human Rights. As such countries recognise marriages performed in other jurisdictions, sometimes extending to same-sex marriage, e.g., France recognising same-sex marriages performed in the Netherlands. I'm no scholar of international law, but AFAIK, a Civil Partnership conducted here does not confer a similar status to PACS whilst in France.
Aaaaanyway, this ended up longer, and slightly more rambly thatn I intended, but in summary - marriage does not 'belong' to any religion, and don't let anyone tell you that it does.

Also, CPs have partnership rights in this country, but the video is about America, which is a whole other kettle of fish.