Since this is a legal question, the exact answer depends on your jurisdiction and the terms of the relevant offer. If necessary, you should seek legal advice from an employment lawyer in your jurisdiction. Below I will give some basic information that applies in most jurisdictions.
The typical legal position in such cases is that a binding employment contract exists once there is an offer and acceptance (with consideration and an intent to create legal relations), which appears to have occurred here. An exception to this can occur if an essential term is not yet agreed, in which case courts may find that the agreement is not sufficiently complete to be legally binding. However, my understanding is that courts have typically viewed employment contracts as binding even if the start date has not yet been agreed (and have usually imposed some implied term here). In cases where an employment offer is conditional on background checks, etc., it may be a "conditional offer" which gives rise to a slightly different kind of contract (see related answer here). From what you have described, it sounds extremely likely that you have a binding contractual employment relationship with this university(at least conditional on background checks) that gives rise to some employment entitlements to you.
Since it is likely that there is already a binding contract here, the university cannot just rescind its offer. Assuming your background check does not raise problems, if the university does not want to follow through on the role then it will need to terminate your employment in accordance with the relevant rules for termination. If this occurs then you may have a valid cause of action for breach of contract, with damages taking account of lost opportunity (e.g., turning down another offer). Laws relating to termination of employment contracts vary significantly by jurisdiction, in terms of allowable causes and notice for termination, and entitlements/damage that may accrue on termination.
As some practical advice here, continue to correspond with the university by email to make it clear that you expect to start in August, and that any delays in starting will have an adverse financial impact. This will put the university on written notice of the adverse effects of its delays, which will put you in a stronger position if this later escalates into a legal dispute. If things get to a point where you are really concerned, go and see an employment lawyer to get advice.