Should we stop using male and female restrooms? Just have a single room for all?
The reason we should keep bathrooms separate is because sometimes females need a place to escape guys and "powder our noses." What does it mean to "powder our noses?" It means if we are "with" a member of the opposite sex, we can resort to the ladies room to chat with a friend about something personal, or just get away from a guy we like too much (to compose ourselves) or get away from a guy we don't like enough (to figure out how to let him down easy. This me seem like a petty reason to keep bathrooms separate, but many females made important decisions about the company they were in while in the sanctity of a ladies room. A more serious reason would be that a mom of a fussy infant needs a place to breastfeed in an environment with those of the same sex (or identify as such). Ladies rooms are more than spaces to poop or pee. We adjust our clothing, hair, faces, attitudes and children in them. We wouldn't be better off without separate bathrooms for the reasons that I sited.
As far as men's rooms is concerned, I have no idea what the ettiquette is like in one. I know I wouldn't feel comfortable watching men's backs leaning in to pee, knowing their hands were on their penis's. What about couples who are dating but haven't "done it"? Wouldn't a female be tempted to peek to know if she was interested in what he has to offer?
Having separate bathrooms may be a remedy for women who really have to "go", but it would also make guys have to wait longer.
If establishments are going to go through all the trouble of tearing down walls to make a single bathroom facility, they'd be better off adding a single toilet bathroom that is unisex. This bathroom would be able to be used by anyone - singly, and with a door that locks, so sexual assault wouldn't be an issue in one.
Besides that fact that your friend is exasperating a controversial topic, transgender individuals make up an extremely small portion of the general population. Americans with disabilities make up a much larger percentage of the population, than individuals who identify as transgender. and There are laws in place to ensure that individuals with disabilities have accommodations in public places. The cost to businesses must have been considerable when they needed to update their facilities to accommodate persons in wheelchairs or with impaired mobility. The amount that it would cost businesses to remodel bathrooms into unisex facilities with multi stalls would be astronomical. Page 29 shows typical costs of remodeling public bathrooms. This would put a financial strain on businesses, when there are no laws that make it mandatory and also the idea of unisex bathrooms don’t have the wide support of the public, like the accommodations for those who are disabled have had.
My solution of adding one unisex bathroom who could accommodate anyone would be more cost effective and logical to serve a segment of the population that is an extremely small percentage of the population. 
"The fact that they let him use the women's room shows that we may be sharing the room with men, at least in Target, and we cannot do anything about it," is not a scientific study. I would need much more evidence than your male friend entering a Target Women's room to convince me that this the norm on a wide spread basis.
The resolution of this debate is: "Should we stop using male and female restrooms? Just have a single room for all?" Single means ONE, not just changing the signs on Men's and Women's rooms. Had you posted the idea of changing the signs in the resolution or in R1 of the debate, I would not have accepted the challenge. Changing the resolution in the third and final round is not ethical.
I will address your argument for the sake of being a good sport, and speculate that if business owners merely changed the signs, genders would separate on their own for the most part, and give "looks" to anyone who "appeared" to be a member of the opposite sex for using the opposite bathroom of what they appear.
Even if your idea of changing the signs was viable, the cost would still be considerably higher than adding a single stall bathroom. Stalls would need to be added around all the urinals or women and girls could cite men for indecent exposure if their genitals were in view. Cost of adding stalls: 3 stalls on sale = $2,517.76 [http://www.sustainablesupply.com...]
Cost of a commercial urinal: $639 - 956 for the urinal alone, not to mention the cost of the additional plumbing, installation and wall repair. [http://www.faucetdirect.com...]
639 x 2 = $1,278....956 x 2 = $1,912 ...so it would cost $1,278 to 1,912 for 2 commercial urinals not including additional plumbing, installation or demolition/re-construction.
I'm sorry that trans gendered people don't have an easy solution to using the bathroom without being uncomfortable, but changing bathrooms to a single bathroom for everyone (the resolution of this debate) is too expensive for businesses, and changing the signs wouldn't do much, because genders would most likely continue to separate themselves.
My solution of adding a single unisex bathroom with a lock, is the most realistic and cost effective solution to the problem. My solution would even offer an added benefit: caretakers of disabled people of the opposite sex would be able to assist them with their bathroom needs privately. Cost of a commercial toilet: $318 - $864 / cost of a commercial sink $126 - 290 not to mention the cost of additional plumbing, installation and construction, but still less than even just adding commercial stalls around urinals.
This link shows how much planning goes into designing or remodeling bathrooms. http://www.inpicta.com...