There's another good answer talking specifics. I'd like to address this part:
Since the United States is a democracy, I thought the reasonable
process and logic should be that POTUS first needs an approval from
the Congress before he could take such an important action.
Congress is the legislative branch, while the Presidency is head of the executive branch. The typical legislative MO is for Congress to pass a law instituting a program, and then put some part of the Executive branch (for example, the Department of Education) in charge of actually managing the program.
So at this point the issue becomes what authority Congress gave that part of the Executive branch when they wrote that law (or often those sets of laws). Often, by design a lot of latitude is given to the Executive. For example, the US State Department is charged by Congress with keeping a list of State Sponsors of Terrorism. There are laws covering what things a country might do to get themselves put on that list, and what the repercussions are to a country that gets put on that list. However, it is up to the US State Department who actually gets put on that list. If the Executive branch (under direction of the President) doesn't want an otherwise qualifying nation on that list, they don't go on it. Its up to executive discretion.
This is actually a very helpful part of the laws, as it makes inclusion on the list a very useful bargaining chip when the State Department (or the President's administration) is negotiating with problematic foreign nations.
So the question here isn't one of "democracy", but rather exactly what latitude Congress gave the Executive branch when they wrote the laws allowing for government-backed student loans. If Congress doesn't like the answer to that question, they are of course perfectly free to change it by passing more laws.
As for the specific answer with student loan forgiveness, that's more of a legal matter, but thankfully @MichaelSeifert over on Law SE already has a good answer there for you. The relevant law Congress passed back in 1965 has this passage:
In carrying out the provisions of this part, the Secretary is
authorized— ...
(2) to enforce, pay, compromise, waive, or release
any right, title, claim, lien, or demand, however acquired, including
any equity or any right of redemption;
So it sure looks like Congress likely already gave the executive branch (specifically the Secretary of Education, who answers to the POTUS but was confirmed by Congress) the power to "waive or release" student loan repayment obligations created under that program. If this Congress didn't want the SoE having that power any more, the surest way to prevent it would have been to change the legislation to take it away.