No, you cannot sue
When you signed up to Uber you agreed to their Terms and Conditions. In Section 6, you agreed that disputes should be settled first by Mediation and, if that fails, by binding Arbitration - both under the respective International Chamber of Commerce Rules. These will be have their seat in Amsterdam and be conducted under Dutch law. This does not mean that you will need to travel to Amsterdam - you will be able to participate remotely.
Notwithstanding, the Australian Consumer Law (ACL) cannot be excluded by contract or choice of law clauses and will still apply. Of direct relevance here is the consumer guarantee (Part 3-2) that services will be provided with "reasonable skill and care" and that this is a 'consumer contract' and subject to the unfair contract provisions of Part 2-3.
The dispute resolution clause 6 is not unfair - Australian law supports alternative dispute resolution and a term is not unfair if it is allowed by law.
The Disclaimers; Limitation of Liability; Indemnity Clause 5 is problematical for Uber as, while limitation of liability clauses are not ipso facto 'unfair', when the seek to limit liability for a breach of a consumer guarantee, they must be limited to what can be excluded by s64A. As written, these are far broader and, notwithstanding the savings clause about 'applicable law' they would probably be held to be misleading. A misleading term is ipso facto unfair. You may be able to approach the court to have Clause 5 declared an unfair term under s250 before proceeding to arbitration - the court may defer that decision to the arbitrator of course.
You can, of course, sue the driver. Uber maintains that their drivers are contractors and not employees (which may or may not be true). If Uber is right, you have no contract with the driver and are free to bring a negligence suit.
