No. Malpractice insurance is specific to clinical care.
Any plan for medical research involving human subjects is examined with a fine tooth comb by the Institutional Review Board to minimize risk and avoid ethical breach, and trial participants sign a detailed consent form. In an industry sponsored trial there is language in the contract to the effect that the sponsor will be responsible for unforeseen medical care costs that are a direct result of trial participation.
Academic institutions keep armies of in house lawyers to manage legal risk and minimize lawsuit damage. If there were potential to be sued by a trial participant the PI and dept chair would work out a plan with Risk Management.