Same-sex marriage is not actually legal in Vermont. Civil unions are, but they're only recognized under Vermont state law, not the Federal government. It would be impossible for Mr. Wick to get citizenship by "marrying" Drew.
It doesn't stop there. In "The Engagement," one of the questions Oswald asks Kate (during the competition between Drew and him to see who is more compatible with his girlfriend) is about how many children she wants. I can't remember if she replies one or two (she changes her answer at one point), but it sure wasn't zero. Plus, the entire episode "Kate's Family" centered on her desire to have children. She becomes so enamored with the neighbor kids that she is babysitting that she starts to think of them as her own and tries to trick them into staying with her after their mother returns. At the end, she complains to Drew about how much she wants a life with a nice, supportive guy and a couple of kids. Maybe it could be argued that this is different, because she was actually in a position of thinking she was pregnant. However, if she always wanted to have kids in the past and now has found the love of her life, why would she change her mind now? If the writers couldn't think of a more plausible reason to break up Drew and Kate than this, then maybe they shouldn't have done it.
Kate says she doesn't want to have kids, but in "Drew Live," she tells Lewis she'd be willing to make that sacrifice: "9 months without beer, but after that I'd have a little drinking buddy."
(After Drew walks into Mr. Wick's office)
Mr. Wick: Good god, Carey; you're fired and you've still got a better attendence than the other employees.
Drew (to Kate): Do you know where my pee bottle is?
Mr. Wick (to Drew): You zig and zag, but your breath is on my fluffy cotton tail.
Drew (after Speedy tears the head off the doll): I hope a real baby has a stronger neck.
Drew (talking about what Kate would be like pregnant): Wow, her boobs are gonna be huge!
Kate (talking about the benefits of not having any children): We're never gonna have to see any shows "on ice."
Drew (to Mr. Wick): You know, someday I will be trapped in a loveless, sexless marriage--but it's not gonna be with YOU.
Mr. Wick: Is it so hard to love a bankrupt, middle-aged English-man with one testicle?
Drew: Well, you are in America--the Land of Meat and Potatoes, not Meat and PotatO.
(Mr. Wick needs a green card marriage.)
Mr. Wick: I can't get married! All women I know hate me!
Drew: Don't forget the fellas.
(Drew gets up to leave.)
Mr. Wick: Wait a second. I did forget the fellas. All I have to do is marry a man!
Drew: You can't marry a man.
Mr. Wick: Yes, I can! There's a state where it's legal now! (Gets down on one knee) Drew, will you marry me in Vermont, the state that makes New Hampshire nervous?
The Go-Go's appear as themselves and sing the theme.