I don't doubt for a second that June wants marriage.
I just don't think that any pressure will motivate Greg to actually start planning a wedding. Why should he? He got her to move significantly closer, she's given into his "BDSM" lifestyle (let's face it-- the BDSM dynamic is less of a kink than it is Greg being a controlling and manipulative pig about the relationship in general), and she worships him publically all of the time. June has pretty much given up everything without needing a marriage to do so ... What motivation does Greg have to get married now?
Also, if he marries June, that would be his third marriage? I thought he was only married once before? Jeez, if anybody has been previously married more than once before they've even hit 35-40, that waves some red fucking flags.
The only thing that could convince Greg, in my opinion, is if she actually grows a spine and thinks for herself by not agreeing to be in his videos anymore. We all know that Greg's audience would dwindle without Shoe in a lot of his videos. If the growth of his Star Wars and Joker toy collection was put to a halt, I think he'd actually consider marrying her soon.