The answer is no, the resulting sequence wouldn't be any more random than any other sequence the generator is capable of producing.
Pseudo random number generators produce a sequence of values algorithmically based on some internal state. Eventually (which may be a very long time for a good PRNG!) the generator will end up in a state it has visited before because there are a finite number of states. Everything from that point on will repeat identically, since each state will inevitably lead to the same subsequent state when cranked through a deterministic algorithm. In other words, PRNGs all produce a sequence of values which eventually cycles. Java's PRNG cycles in about 248 iterations. Mersenne Twister's cycle length is about 219937—you'll never cycle through its entire state space in your lifetime, but it's still producing a deterministic sequence of values.
While the details vary from PRNG to PRNG, seeding is used to determine the initial state. Choosing a truly random seed value means that you are climbing into the deterministic cycle at a randomly selected point, but it remains a deterministic sequence from that point forward.