No, you do not.
You will need an NIE number though, and once you have a contract, your employer should register that with the Employment Agency (SEPE/INEM) and you will get a social security number.
Some employers will ask you to arrange for this number yourself. Info in this same section of the FAQ´s Forum on how to do so.
Autonomos (self employed) may wish to use a gestor de contabilidad to help them set up and carry out their trimester IVA returns and annual taxes.
Once you have your contract, you can register with the Seguridad Social (INSS) for healthcare and register on the Registro de Extranjeros as well, to formalise your status as resident then. Depending on the contract and your wages, the contract will both serve you to meet the requisites re ´proof of healthcare´ and ´proof of income´.