That's not how the criminal justice system works. You file a criminal report (ideally not 30 years late), and you provide the appropriate law enforcement agency with as much information and evidence as you can provide. The standard protocol for filing a criminal report is making an official statement. If there is no report, no statement, no evidence, there is no investigation.
Except that this isn't a criminal proceeding. You do know the FBI does background investigations on someone being nominated to the highest court in the land right? You do know that many times the FBI has re-opened background investigations based on new allegations right? Specifically the FBI re-opened the background investigation into Clarence Thomas when Anita Hill came forward with her allegations, which again in an ironic twist had senate republicans demanding an FBI investigation into the new claims (some of those senators still sit on the committee today). That investigation lasted 2 days.
You don't get to demand the FBI go dig through some ones life just based on statements alone. That is their discretion,
Really, digging through someone's life based on statements is literally a major part of a background investigation, they need to be corroborated or exculpated that's literally what an investigation is. Before nominating a person for a lifetime appointment to the highest court in the land this always happens and Kavanaugh has had them before.
That is their discretion,
And the presidents

but obstruction of justice is one of Trumps specialties!!!
BTW as it turns out there is no statute of limitations in Maryland for this alleged crime and no reason why a sitting judge on the SC can't be indicted.
What is a shame is how bad this is being handled by both sides it seems to me American politicians have learned nothing in the 27 years since Anita Hill and both sides of the aisle should be ashamed with their behavior.
On top of that the American people IMO should be furious with their politicians who have created this hyper-politicized environment overall not specifically this noms hearing. It isn't healthy when the goal of each party is their own interests and not the interest of truth and the American people!