Yes I agree with CathL, insurance companies will always try to get around paying for things like this.
What sort of damage is it? Is it something that is visible to the handset? Is it something that prevents you using the phone? I suppose simply providing phone records to demonstrate that you were using your phone prior to going away on vacation would not necessarily prove that you were able to use the handset, because you could have put your sim into another phone but it may be worth a shot. I don't know, are there ways that telephone service providers can record what sort of handset you are operating the sim card from? What about pictures? I know its unlikely, but do you have any photographs of the phone lying around in the background?
I would write to them and ask them to stipulate specifically the sort of evidence they require. If they cannot specify what they need, or if is unreasonable for them to expect that you could provide that sort of evidence, I would escalate your claim through the normal appeal procedures of the insurer. All insurers should have in place some avenue to dispute decisions and generally if you make enough of a song and dance about it they may eventually cave.
Once you have tried to resolve the problem through the insurance companies internal dispute resolution procedures, I would then threaten to contact the Financial Ombudsman Service - which also deals with insurance matters.
This is the link
http://www.fos.org.au/resolving-disputes/before-you-lodge-a-dispute/
Good luck!