The Indian calendar is a lunisolar calendar which synchronizes itself with the solar cycle every year even though it is based on the lunar cycle.
Check this website for more calendar information http://webexhibits.org/calendars/calendar.html
Arabs got their basic astronomical knowledge from ancient Greek and Hindu works on that subject. This is known to have happened around 700-1000 A.D., but not before. If I am not mistaken, the Arabic calendar predates Islam and since Islam codified and rigidified the practice of religion as it came into being, there wasn’t much scope for alteration later on.
Pope Greogary reformed the “Christian” calendar when the errors of Julian calendar became too great, but I don’t think it is possible for Islamic calendar to have the same reformation, as any change would be the invalidation of the word of GOD.