What would be the implications for complexity theory if you could compute the Kolmogorov complexity of a string generated by a psuedorandom generator?
If we're talking about a generator who can handle any length $n$ seed (perhaps this is more cryptographic PRG oriented), and stretch it to some length $n'>n$ pseudorandom string, then the answer is no. The reason actually has nothing to do with the properties of PRGs, but simply relies on the fact that the output of the generator is computable, and that its range is infinite.
Kolmogorov's complexity isn't computable on any infinite recursively enumerable set of strings. To show this you can follow the standard proof of uncomputability of Kolmogorov's complexity. Since the set is infinite, it contains strings of arbitrarily high Kolmogorov's complexity, so you can write a program which enumerates them until it finds some string of high enough complexity, and then stop and output it. This was also answered in this math.se question.