Dine, you can do that, no problem. Only, that would be horribly confusing: imaging starting your day at 20:00 hours somewhere in June, because you’re shifting each day a little (and the Sun would rise, say, at 16:00 hours). We live, as practically any organism on this planet, by the day-night cyclus, so that’s what we’ll have to do with. Since we also live by the seasons, there’s also a year involved; and those two just don’t add up in whole numbers.
Actually, Nature would tend to fix this, that is, making the yearly orbit a whole number of days. Mercury has this behaviour, where 3 rotations around its axis equal 2 rotations of the planet around the Sun. The reason this doesn’t happen on Earth are several; most importantly 1/ the Moon is disturbing this resonance effect, and 2/ the Earth is quite a bit further from the main centre of mass, the Sun, that contributes to this.
Now that I’m on this, there is another thing to point out, just to confuse things further: one day on Earth (from Sun in the South to Sun in the South; 24 hours exactly) is not the same as a rotation around the Earth’s axis. Since the Earth moves a bit in its orbit in the meantime, it’s need to rotate a bit extra to align with the Sun again: 4 minutes extra. So a day (Sun to Sun) is 24*60=1440 minutes, but a revolution of the Earth 1436.
If you think this carefully through, you’ll see it has nothing to do with the leap year though. It’s just another inconvenience, mostly for astronomers: to them, the night sky appears to drift every day with 4 minutes (as you will notice yourselves, since different constallations appear at different points in the year, but at the same position on the same time). This 1436 minute day is called a siderial (“star”) day.
When I mentioned the orbit/axis rotation resonance for Mercury, I was mentioning the actual rotation (siderial day). So you wouldn’t really notice it when you’d be standing on Mercury.
@bpeoples: that was my last point. 12 months & 12 zodiac signs equal 12 (well, roughly, but hey) Moons in a year.