There is a law, the First Amendment, which tends to support the right of a university employee to use an electronic soap box to speak freely, which right includes commercial speech. Congress has not passed any law that implies that advertising jobs by university emails is prohibited. The prospects for such a law being passed and surviving judicial review are quite slim. Since the law would infringe on a fundamental right, the law would have to pass strict scrutiny, and I see no way in which such a law would address a compelling government interest.
At the state level, the answer would depend in part on the constitution of the state and what actual laws were passed, and I don't know of any state where a law was passed prohibiting the use of university emails to disseminate messages of a certain type. A particular university might, however, find it to be in its interest to limit how the university email system can be used. Here is one example of a "responsible use" policy. The following policy statement comes close to what you are looking for:
Refrain from using those resources for personal commercial purposes or for personal financial or other gain. Personal use of university
computing resources for other purposes is permitted when it does not
consume a significant amount of those resources, does not interfere
with the performance of the user's job or other university
responsibilities, and is otherwise in compliance with this policy.
Further limits may be imposed upon personal use in accordance with
normal supervisory procedures or as a matter of college, department,
or unit policy.
However, first, this only applies to personal gain, therefore I can send email announcing that there is a job available or that a person is available for a job (this happens all the time, especially the former). The question of whether sending an email to a couple of friends consumes a significant amount of university resources has to be judged on a case by case basis, but unless one is using university email to launch millions of emails, the resources consumed in processing ordinary non-mass email is not significant. There is, you will notice, an option for college, department or unit imposition of rules. It is a fact that some departments had and may still have (vague) policies against "commercial use of department email", but again it depends on what you mean by "commercial use". In other words, it depends on the university, or department.