Falsely Accused of Cheating on a College Final
Most universities will have some sort of quasi-judicial appeal process.  You can try to navigate it yourself, but I would strongly suggest that you find an expert to help you.  Some universities may have an ombudsman or some sort of "public defender" service that provides trained students you can work with.  You could also talk to students or faculty who have experience with the process, or even consider hiring a lawyer.

Part of what you'll want to know is who will judge such a case, what sort of evidence  you can present, and what "standard of proof" is expected.  

Some evidence that you could consider trying to present, if possible:


Expert testimony from an impartial person familiar with the online system, to explain that a logged access does not necessarily prove that you actually looked at the information, and to explain how it could happen without your knowledge.  For instance, you might be able to get someone from the university's IT office to testify, or a computer science professor.  (People who work in technical fields are often pretty sensitive to cases where a poor understanding of technology by those in authority results in an injustice, so I think you might not have too hard a time recruiting someone.) 
Testimony from other students that they did not see you look at your phone or any other device during the exam
Testimony from the professor as to whether she saw you look at your phone, etc


As for your classes for next quarter, it would be worth it to talk to the instructors of those classes.  You can tell them that you were charged with dishonesty and you are appealing, but that in any case you have learned the course material.  They will typically have the authority to waive the prerequisite requirement, so that you can take their course despite not having passed the previous course, and they might be willing to agree that it serves no purpose to delay your progress through the program by making you retake a course whose material you already learned.  This way you can continue to move through your courses even if your appeal is not resolved before the next term starts.