I don’t know if I’d go with gender equality notions per se, after all Ned is still the one who says this:

Arya cocked her head to one side. “Can I be a king’s councillor and build castles and become the High Septon?”

“You,” Ned said, kissing her lightly on the brow, “will marry a king and rule his castle, and your sons will be knights and princes and lords and, yes, perhaps even a High Septon.”

Arya screwed up her face. “No,” she said, “that’s Sansa.” She folded up her right leg and resumed her balancing. Ned sighed and left her there.

I think the thing with Ned is that he cannot help but see Lyanna in Arya’s face, and so he can’t bring himself to outright forbid Arya to pursue those things she has in common with Lyanna. It’s a personal thing, founded in his specific relationship with Lyanna, as opposed to a general commitment to gender equality as a concept. 

Squaring the circle between Ned’s allowing Arya to have a “dancing master” and telling her that she can’t follow a male-gendered “career” path is that Ned is also someone profoundly marked by his past: 

“It has a name, does it?” Her father sighed. “Ah, Arya. You have a wildness in you, child. ‘The wolf blood,’ my father used to call it. Lyanna had a touch of it, and my brother Brandon more than a touch. It brought them both to an early grave.”

“Lyanna was beautiful,” Arya said, startled. Everybody said so. It was not a thing that was ever said of Arya.

“She was,” Eddard Stark agreed, “beautiful, and willful, and dead before her time.”

So while Ned can’t bring himself to stop Arya from mimicing Lyanna, he also feels strongly that Lyanna’s “wolf’s blood,” her “willful” nature lead to her early death and doesn’t want Arya to go down that path. Thus, he’ll allow her indulgences like swordfighting training, but he wants her to be safely married and live a long and conventional life.