In order to turn the universe into a solution, one would need to combine all of the available oxygen with the hydrogen in order to make water. (This isn't actually possible, but the question is hypothetical anyway.)
Look at the abundance of elements in the universe: http://en.wikipedia.org/wiki/Abundance_of_the_chemical_elements
Because of the huge amount of hydrogen compared to anything else, we can safely conclude that 100% of the water molecules would be hydronium (H3O+), which gives us a ratio of 1. The pH value is the negative log10 of that ratio, or -log10(1) in this case.
The pH of the universe (were it to be made into a solution) is 0, so.
I accept that there may be buffering due to the other minor elements present, but the preponderance of ionic hydrogen easily counters that; if anything, it would make the pH negative.