Short answer: no. The vernacular for mens rugby is just "rugby." The language hasn't evolved beyond that at this point. It's kind of an "it is what it is" thing right now.

You can dislike the gendered language as it's not particularly inclusive, but if you watch much rugby you will see that they are effectively different sports. It's like tennis. Even Serena Williams said very clearly that mens and womens tennis are different sports due to the physiological differences between men and women in sports where strength is required.

BTW: I'm not advocating for anything with respect to what language we use. It just is how we refer to the sports at this point.

Edit: to expand a little, should we call 7s just "rugby" or should we differentiate because it's so different than traditional rugby union? What about rugby league? 10s? Etc. Is Ruby Tui wrong for referring to her sport as "womens rugby"?

I don't think "men's rugby", as you call it, is gendered. I think it's open to all.

The biggest and best version of something tends to just be called that thing with no qualifiers. Like "the world cup" refers to the mens football world cup unless otherwise specified