Well they already tried suing them when they began accepting girls and changed their name the first time to Scouts BSA, but that didn't work. But truthfully the two organizations have different missions and methods.
Historically, a lot of girls who joined GSUSA thinking it was going to be Boy Scouts for girls were disappointed and would leave. GSUSA is more about empowering girls through community engagement and exploring careers. Yes there can also be camping, hiking, etc but these are more or less optional components, up to each troop to integrate. Rank advancement is based on age and grade level, while awards are based on merit.
Boy Scouts is much more focused on outdoor skills and citizenship. These are integrated into the program in that advancement in rank is based solely on merit and demonstrating proficiency with these skills. You can spend six years in Boy Scouts and never make it past Tenderfoot.
So for girls who want more emphasis on the outdoors built into their program, Scouting America would be the better option. For those that want more flexibility and are less outdoorsy, GSUSA is still an option. Both are good programs. I have kids in both. There are some things I like better about GSUSA and some I like more in Boy Scouts. I think Boy Scouts is a more challenging program overall, but GSUSA's Gold Award is way more challenging to achieve than an Eagle project. I definitely prefer GSUSA not having a religious requirement.
Both programs will continue to adapt and change. Both have been experiencing declines in membership for decades anyway, so there's bigger problems that they're facing.
I loved being part of the Boys Scouts of America. Great experience all around and went all the way to Eagle Scout. Like lots of big organizations it all depends on the local chapter and has some issue through the system. Hopefully the name change gives a chance for it to get better.