No. A court booking can't be turned into a match, or a match into a court booking. They follow different rules (payment, level range, gender), so you need to cancel one and create the other.
Workaround
Go to Calendar → Schedule and open the booking or match.
Click the cancel button at the bottom, for example Cancel court booking or Cancel public match.
For each player, choose the refund. Full refund is pre-selected for every player who can be refunded, so check each player before you confirm.
Create the new booking or match on the same court and time.
Add the same players. If they have already paid and you didn't refund them, set their payment to Marked as paid.
Private match to public match
The player who created a split-payment private match can turn it into a public match in the app with Turn into a public match. This can't be undone, and there is no option for this in the admin.
Good to know
Players get cancellation and booking emails when you cancel and recreate.
See "How do I cancel a booking and refund players?" for the refund options.