Spay is the generally-accepted term used to describe the surgical procedure referring to reproductive alteration of a female dog or cat.

Neuter is the generally-accepted term used to describe the surgical procedure referring to reproductive alteration of a male dog or cat.
In fact, the two terms are interchangeable.
Why?
“Spay” comes from the Greek word spathe, meaning “broad blade.” Apparently, the use of a blade (scalpel) to alter pets got conveyed to mean the procedure itself.
“Neuter” comes from the Latin neuter, literally meaning “neither one nor the other.” In the beginning of the last century its use applied to male and female cats, but not to dogs.
Word usage has never been concrete. Languages, all languages, change with time. Technically, a male dog can be spayed and a female dog can be neutered. Someday, however, etymology entries will label that usage as “archaic.”
Terms that are gender-specific and survive the test of time are orchiectomy and ovariohysterectomy. Furthermore, they apply to all mammalian species.
Orchiectomy comes from the Greek word orkhis, meaning “testicle*.” The suffix comes from the Greek term ektomia meaning “a cutting out.” Therefore, orchiectomy is the removal of the testicles by cutting.
Ovariohysterectomy is the term describing surgical removal of the ovaries and uterus. “Ovary” comes from the Latin word ovum, meaning “egg.”
“Hysterectomy” comes from the Latin term hystera, meaning “womb.” Don’t be mad at me, but the name comes from the pre-1900s concept that women’s hysterical behavior came from their uteri. Hysterectomy, then, would be expected to remove their hysteria. You can’t make this stuff up.
The related procedure, oophorectomy, or ovariectomy, removes only the ovaries. The Greek root oophoros means egg-bearing. Therefore, removing ovaries removes the ability to “bear eggs.”
Want a single, simple, all-inclusive term? Gonadectomy. From the Latin gonas, meaning “seed,” it covers removing the “seeds” of future generations from any species of any gender.
Can’t you just hear Bob Barker say, “Have your pets gonadectomized!”?
See you tomorrow, Dr. Randolph.
*The orchid root and the testicle share a shape. The testicle was named first and the orchid first discovered in 1798, inherited the Greek name.