to be relentlessly gay means not to buy into the "i don't care what they do in the bedroom but keep it there" crap purveyed by people who don't mind pinching women on the butt in public and are not offended by straight couples holding hands as they walk down the sidewalk or straight couples kissing in the doorway as one of them goes off to work. it means not believing that you have to suppress who you are so that bigots don't get all wobbly inside. it means being able to have a picture of your husband or wife on your desk at work even if the spouse is the same gender as you. it means being able to SAY "my husband" or "my wife" in casual conversation, even if said spouse is the same gender as you, just as if it's a normal thing -- which it IS -- without fearing that some homophobe will piss his diapers at the very thought, even though he has pictures of HIS wife on HIS desk. it means being butch or femme or just an ordinary guy or gal as your personality or mood of the moment suits you without caring if some idiot swallows his or her chewing gum at the very sight, sound or idea of you. being relentlessly yourself, whether gay or straight, is a good thing. the next time your friend is called that, the appropriate response is a big smile and 'oh, THANK you!"

g

Might be a reference to this.

[snopes.com]

Either way it is probably someone being an asshole.

It means she’s an idiot and sees this;

Reminds me of this thing going round the internet a while back where someone wrote a letter complaining that their neighbour's rainbow-festooned house and lawn were 'relentlessly gay.'

I think it may be similar to "uppity black woman". In other words, it is a way for someone to say that you should know your place and not be so outwardly expressive of who you are. They say that to shut black women up. They say that to tell you to tone down your gay. Both expressions are insulting. Tell you friend. "You be You".