Professorship without PhD in the United States

At most US institutions, you need the terminal degree in your field. For many fields this is a PhD, but in some it might be an EdD, a DMA, a DPH, a ThD, etc. However, all of these degrees are considered to one level or another to be research doctorates. Presumably faculty positions at a medical school require an MD or equivalent. Likewise for other professional schools. Positions in the visual arts, theater, dance, creative writing, cinematography, etc., may only require the MFA (Master of Fine Arts) degree, as it is considered to be the terminal degree in those fields.

"Regulations" about the requirements to hold certain academic ranks and perform certain academic duties (like mentor graduate students) are made at the institutional level.

I was thrown out of a small mid-western college (not a university) in my first year. I was persuaded to apply to Harvard about 8 years later. I took the exams and was admitted. I paid my way by working a 40 hr/week full-time job, admittedly against the rules, at an electronics firm throughout the 4 years. I graduated with a decent, but not outstanding A.B. in a scientific field, and had, incidentally, become Chief Electronics Engineer at the firm where I had been employed. This was followed by employment in diverse research environments, then in a think-tank in Cambridge and, finally with two offers of tenured full professorship at major universities. I took one, and thus became a full professor at a major university without ever having taken a course in graduate school or having had any prior appointment as Assistant or Associate Professor. I am currently Professor Emeritus and continue to direct PhD candidates. The answer to the question is, therefore, YES

Andrew Casson never completed his Ph.D. and is a Professor at Yale University (and was previously a Professor at UT Austin and UC Berkeley).