No. It would not be a written contract unless the parties signed it.
However, Florida law will enforce oral contracts in certain situations. In addition, there are other legal theories that can be used to recover such as "promissory estoppel" and "unjust enrichment".
In your case, the dispute will resolve around what either of you state was the understanding. The contractor will be at a disadvantage, particularly in the license complaint environment, because Florida best practices
are for contractors to provide written, detailed contracts, which your contractor failed to do.
In addition, the invoice he provided would not be admissible as evidence as to what was agreed because he unilaterally changed it (e.g. scope of work/contract).
So, the most that exists is an oral contract with some invoices that could be used to determine what work was agreed to. All the extra stuff the contractor now asserts he is owed for (e.g. 20% fee on top of labor) may not be recoverable by him unless that was verbally explained to you at the begining of the relationship.