This may be a more detailed answer than you want, but here's essentially how it works. The court in which A-Rod has filed his papers is a "district court," which means it's a trial level court. But that's confusing here. Trial courts usually hold trials, with juries, and find facts, and issue verdicts. This is not that kind of case. Sometimes, trial level courts actually act like appellate courts -- that is, they review what another court or tribunal (in this case an arbitrator) has done.
When you're a reviewing court, your job is usually quite different from the original court. It's not (at least usually not) a "do over." Instead what it is is a chance for the loser in the lower court to come to you and say, "hey this was wrong, and here's why." That's what A-Rod has done here. The papers he has filed seek to "vacate" the arbitrators' award.
Depending on the kind of case and how it was decided, reviewing courts don't attempt to go back and re-do everything the lower court did. You have to pick your issues and say here are a couple of points where I think a mistake was made. Depending on what your arguments are, and the way the lower court decided things, the reviewing court will give certain deference to the lower court. (The level of deference and the justifications for deference are the subject of a lot of law that has developed over a long time.) So, for example, a reviewing court will almost never second guess a lower court's or arbitrator's view of credibility of a witness -- because the arbitrator actually got to sit in the same room with a witness and maybe see him sweat or something, etc. In reviews from arbitrations, the level of review is very deferential -- on the ground that arbitration is supposed to be an alternative to court and if we're just going to retry everything in court why bother.
So, that's a long intro to the answer to your question. When a court is engaging in deferential review, it almost always takes no new evidence. The question is not "what are the facts"? The questions is "based on what was given to the arbitrator, was his decision reasonable, even if other people might disagree"? It would hardly be fair to reverse an arbtirator -- say "you got this wrong" -- by considering evidence that the arbitrator didn't even have a chance to consider. So, all the district court is going to do here -- most likely -- is look at the evidence that already was presented (documents and transcripts of witnesses) and decide whether a reasonable person looking at that evidence might do what the arbitrator did. Even if the judge thinks he would have done something different, that's not the standard. He has to affirm. So, the only evidence that MLB will produce here are the pieces that they already provided in arbitration to show the court that there was enough to sustain the arbitrator's award. In an appeal context, you can't give the judge 10,000 pieces of paper, so you usually end up fighting over just a few key documents or bits of testimony; thus the notion that any new evidence that would make a prosecutor take notice might come out is very unlikley.