Why does most of the world celebrate new years on January 1st?

Because the calendar most in use today is the Gregorian calendar and New Year's Day has been defined as January 1.

New Year's Day is a definition put in place and not a natural phenomenon. For instance, the ancient Egyptians had the first day of Thoth as new years day; this was at the reappearance of Sirius at a particular point in the sky. They also used a calendar 365.25 days, more than 2000 years before Caesar introduced it in Rome. After stabilization the 1st of Thoth falls on what we call 29th August.

On 14 September 1580 a commission set up by Pope Gregory XIII, to reform the calendar, signed its official report. In that report was a clause to standardize new year as January 1, the date used by the Julian calendar. This was implemented in October 1582.

This does not mean there are no other new year's days in use. Any calendar not based on the Gregorian calendar will have a different new year. And they are all correct.

New Year is an arbitrary day, introduced by man for the purposes of man.