Car accident with no police involvement and no insurance; is there any real evidence against me / will I have to pay the damage?

You should not be driving without insurance. If she reports you, it would not be good, as you'd likely get cited and have to go to court. Best bet is to pay her for the damage you caused (have her sign a release), and get insurance ASAP.

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Yes you could be liable for the damage to her car and without insurance the repairs may be your financial responsibility, there is a strong possibility that an insurance claim will be filed and when it is found out that you don't have insurance your license will likely be suspended. A driver's license is a privileged and can be suspended if you drive without insurance. A driver's license does not mean you can drive a an uninsured vehicle. Who do you think should bear your responsibility for insuring the public against your irresponsibility.

Sorry, no sympathy here. Get insurance and be a responsible driver. I would bet 10 to 1 that you continue to drive without insurance.

Good luck.

Legal disclaimer: I am not your attorney and we have no attorney/client relationship. This response is submitted for informational purposes only. This is not legal advice, nor is it intended to be taken as such. Anyone considering the above referenced response and should always consult directly with an attorney within your jurisdiction before taking any action based on this, or any other information. This information is provided without any cost and therefore is worth what you paid for it. Again this information does not create any privileged attorney/client relationship and you are cautioned about information you convey to any person in a public forum considering your case, especially if it is criminal in nature. If you would like to consult with me you may contact me at (216)316-3161. I am admitted to practice in the State of Ohio and the State of Michigan.

Since you are writing from Grand Rapids, Michigan, I assume the crash happened in Michigan and if so then you as an uninsured driver are not protected by Michigan's mini tort law and you are now exposed to the total cost of the vehicle damage.
Also, you are now personally liable for any injuries if the other driver is seriously injured, and you will likely be sued by the other driver's auto insurance company if she recovers after bringing an uninsured motorist claim against her insurance company. http://www.michiganautolaw.com/nofault/un-motorist-coverage.php
The penalties for driving uninsured in Michigan are extremely harsh.
Yes, there is real evidence against you - as you say you were backing out of your driveway and hit a vehicle with the right of way at an intersection that was waiting for a stoplight.
Finally, I would not make the leap that because the vehicle suffered minimal visible damage, that this would mean the occupant is fine. Vehicle damage does not correlate with occupant injury, and there are hundreds of cases every year where the vehicle is fine, but the occupant suffers severe injury or even death. Think of Sam Kinison or Natasha Richardson, as two examples of why pictures of a car cannot rule out injury to an occupant.