Nope. Earth's rotation doesn't govern time.
Time is a human construct and is governed by human imagination.
(not to be confused with time as understood in "space-time")
Time has been measured using the properties of an chemical element called caesium.
It has been so since the sixties and the first caesium clocks.
Before that, time was measured using pendulums.
If earth's rotation governed the march of time, then spinning earth the other way around would make us travel back in time and spinning it faster would propel us to the future...