The major key is present by nature in every note that is played. Therefore, it is interpreted as normal behavior, a happy day in our lives, 'cause that's what we expect to happen.
The minor key is opposed to the major key and it's perceived by us (without being aware) as if there was something wrong, hence sadness or restlessness.
When you hear a note, what you're really hearing are vibrations. For each note there is a whole set of vibrations that take in place, and that we don't even perceive.
When you play a low C, you're not hearing only C, but every other harmonic or overtone that belongs to C. That is, going from lower to higher in pitch, the low C being played, then C (octave), then G, then another C, then E, so on and so forth, each time being less the distance between the current overtone and the next one.
As you may have noticed, this first 5 notes (C, C, G, C, E) form the major triad. This means that by nature, the major triad is always present.
The note that results in the minor triad is E flat, which is the 18th overtone in the harmonic series. For this reason, and because the major triad is always there, it results in a contradiction that our human nature understands as sadness, unconformity, and restlessness.
If you want to go deeper into this, I strongly recommend Leonard Bernstein's lectures "The Unanswered Question", in which he engages in a deep and detailed explanation on this and other similar matters.
Here lies musical universality.
Edit: It can be considered universal, as physics have demonstrated its existence in nature and, as the same Bernstein says, the major triad (and even until the pentatonic scale, which would user overtones 5 and 6) can be found in every culture around the world.