Since you have repeatedly reported this man's conduct to the gym, and the gym has not taken remedial action, it is doubtful that you will be able to persuade the gym to cancel his membership. It is possible, but doubtful. Plus, if you do achieve this result, it is possible this man will sue you for interference with his contractual relationship with the gym.
As an alternative, you could have an attorney review your contract with the gym to determine whether the gym's allowing the man to engage in this behavior is a breach of your gym membership contract, or a breach of the gym's duty of good faith and fair dealing with you under your contract. For this reason, and perhaps just to end what seems to be a disruptive and ongoing dispute between members, the gym may agree to let you cancel your own membership. That way you can go to a new gym where you won't have to interact with the man in question.
Florida has an anti-stalking statute, Section 784.048. Based on your description of the events you seem to have potential grounds for filing a criminal complaint against the man in question. For whatever reason, though, the police have been unwilling as yet to make an arrest.
There are also potential common law remedies you could pursue in a civil action against the man who is stalking and harassing you. What those remedies may be would depend on what specifically he is saying and doing. It is possible that if you filed a lawsuit against this man, he may lose interest in harassing you because the risk of not doing so is simply too high. On the other hand, if he is irrationally fixated upon you, a lawsuit is a potential vehicle for him to get closer to you; e.g., seeing you in deposition, seeing you in court, and so on. Some stalkers relish that opportunity.
Many times these type of disputes can be ended simply by having an attorney write a cease and desist letter to the harasser, cautioning that legal action will follow if the behavior continues. However, when you are dealing with people who are potentially irrational, you unfortunately never know what their next course of action will be.
You need to write a certified letter to the gym, asking them to preserve the security videotapes as it is evidence in anticipated litigation. Alternatively, they can provide you with copies so that you can preserve the evidence. If they refuse, when and if you do file a lawsuit you will want to immediately subpoena those records.
If you choose not to take legal action, as a practical solution you may want to change the dates and times when you attend the gym, and try to stay as close to the gym's staff as possible when he is there.
This response is merely a general discussion of an issue based on the information provided. It is not intended as legal advice and does not form an attorney-client relationship. You are encouraged to seek out an attorney of your choosing in your local jurisdiction, and to discuss your legal issue with that attorney.