Step 1: People make the argument "gay marriage cannot exist because marriage is between a man and a woman".
Step 2: Onus is on them to demonstrate their argument.
Step 3: Profit!
Yes, iNow is saying that. He is saying that because those people trying to present it as an argument have failed to fulfil Step 2 above. He is rationally, morally, and legally correct to take a sceptical view of their reasoning.
I am not certain what legal status the Oxford Dictionary of Law has with regards to defining terms, but I somehow doubt that it is referred to in court rooms. I also note that it pertains to UK legislation and do not see how that is relevant to a discussion of legislation in the USA. The origin of your definitions must be consistent with the legal framework you are applying them to, or they are worthless.
I don't think iNow stated this explicitly, but he certainly implied it: that is begging the question.