Actually there is. The 14th amendment guarantees "equal protection under the law." This applies to accusers and defendants.
No it does not actually. Everybody is equal before the law, which means anybody accused of a crime does (or at least should) have the same rights. However, accusers and defendants have different roles and thus are treated differently. A defendant has a right to a jury trial. An accuser does not, unless she becomes a defendant (for making a false accusation for example).
An accuser cannot compel a case to go to trial. All criminal trials are brought not in the name of the accused but in the name of the people. The prosecuting attorney represents the people, not the accuser. As such it is their responsibility to choose which cases to bring to trial and which not, based on evidence and also their impression of the credibility of the accuser.
Interesting legal theory on the 14th amendment though. I suggest you try it on an actual lawyer sometime and watch them laugh.
How the Prosecutor Decides Which Cases to Charge
So yeah, the system is not perfect, but your idea to take every possible criminal complaint to trial would be a disaster. Imagine somebody accuses you of a crime for no reason. Should you have to go through a trial just because a bogus accusation was made?