No, it is not

College Investor: "Is Student Loan Forgiveness By Executive Order Legal?"

The President does not have the legal authority to forgive student loans on his own. Only Congress has the power of the purse. Executive action can be used only when it has been specifically authorized by Congress.

The executive branch cannot spend money that has not been appropriated by Congress, per 31 USC 1301 et seq (Antideficiency Act (P.L. 97-258)) and Article I, Section 7, Clause 7 of the U.S. Constitution.

Know who else came this exact same conclusion? Nancy Pelosi:

Why did Pelosi say “the president can’t do it.” She said that because the president can’t do it. Why did Pelosi say “that’s not even a discussion”? She said that because everybody knows that the president can’t do it. Why did Pelosi say that this was a matter for Congress? She said that because this is a matter for Congress.

The simplest explanation is that Congress decides how to spend money and only Congress can forgive this debt.