Nope. Once a player has an assigned number they can't change it:
"Before the commencement of each Season each Club shall allocate a different shirt number to each member of its first team squad.
A Club shall likewise allocate a shirt number to any Player joining its first team squad during the Season.
Save with the prior written consent of the Board shirt numbers shall commence with the number one and shall be allocated consecutively.
While he remains with the Club a Player will retain his shirt number throughout the Season for which it was allocated.
Upon a Player leaving a Club the shirt number allocated to him may be re-allocated."
Said player James has left the club, number 21 has been re-allocated to Cavani, and Number 7 is now free and been allocated to Ronaldo.
It doesn't say the re-allocated number has to go to a new signing.