Nope. The "Dark Lady" is not a play or a poem, but rather scholars' nickname for the subject of a couple dozen Shakespearian sonnets, and one of the plausible candidates is the woman in Newman's novel, which isn't really "about" Shakespeare anyway.

Here's one, Sonnet 130, where our boy is taking the piss, trying to reframe cultural expectations around literary love, or maybe a bit of both. Poems like this show why he was both brilliant and perfectly likely to have been the educated-ish son of a provincial merchant. Christopher Marlowe didn't write shit like this.

My mistress’ eyes are nothing like the sun;
Coral is far more red, than her lips red:
If snow be white, why then her breasts are dun;
If hairs be wires, black wires grow on her head.
I have seen roses damasked, red and white,
But no such roses see I in her cheeks;
And in some perfumes is there more delight
Than in the breath that from my mistress reeks.
I love to hear her speak, yet well I know
That music hath a far more pleasing sound:
I grant I never saw a goddess go,
My mistress, when she walks, treads on the ground:
And yet by heaven, I think my love as rare,
As any she belied with false compare.