Yes, but unlike me (and apparently @roycroftmom ) you apparently do not have a law degree.
The “penumbra” rationale was used to read a generalized right to privacy into the 4th amendment, going beyond the specific language about searches & seizure – first in contraception cases and then in Roe vs. Wade.
14th Amendment equal protection would not be a way of addressing abortion rights because an anti-abortion law could be written in neutral terms to prevent any “person” from having an abortion.
The “intermediate scrutiny” test that roycroftmom mentioned was established in the case Craig v. Boren (1976)-- which was a case involving a state law that set a different, lower drinking age for women than for men.
The 14th Amendment does not prevent the government from making distinctions among classes of people, but the distinctions need to be justified by good reasons and different types of classifications must meet different tests.
If the classification is tied to a category based on a “suspect” classification, such as race – or if it infringes on a fundamental constitutional right (example: freedom of religion), then the law must pass a “strict scrutiny” test, meaning that a compelling government interest must be shown, and the law must be narrowly tailored to meet that goal. See https://www.law.cornell.edu/wex/strict_scrutiny
Passage of the ERA would have put gender discrimination on the same level.
Instead, the Craig v. Boren case established an “intermediate” scrutiny test for gender-based distinctions – which is invoked when there is a “protected” (rather than “suspect”) class and has also been applied in some first amendment cases. In those cases, the law must further an important government interest and must do so by means that are substantially related to that interest. See https://www.law.cornell.edu/wex/intermediate_scrutiny
There is a third, lesser level of scrutiny called the “rational basis” test, which requires only a showing of a legitimate state interest, and a rational connection between the laws means and goals. https://www.law.cornell.edu/wex/rational_basis_test (This is where things got kind of messy in litigation over gay marriage – courts had a hard time figuring out where that one fit)
In the draft registration case, the district court applied the intermediate (or “heightened”) review standards.
I think legally the rationale is pretty solid and very likely to stand up, but it is only a declaratory relief action - no injunction – so this also is may be addressed by Congress one way or another in the interim. There is no downside to the government for requiring females to register – how is anyone actually harmed by registration? It’s actually the 2nd step – classification – which is the trickier issue – but classification would not take place unless a draft was actually instated.