You still haven't answered.
They are connected, but the relationship works only in one direction. How society values a particular good dictates its price. When that value changes, the price changes with it. But the reverse doesn't work. Changing the price by fiat doesn't change how society values the good.
That's a naive view of the world. People aren't paid based how society values their work. That only sets the ceiling, but not the floor for their wages.
Otherwise programmers would be paid same for same work at same company whether they are based in Silicon Valley or Bangalore, since society doesn't care where the code is created. But if someone transfers from US to India office, they will take a huge pay cut at most companies.
Pay is based on opportunity costs to the company and the employee. Minimum wage makes it so that if a company is not willing to pay it, it will instead incur an opportunity cost of not having a needed position filled, so instead of capturing a smaller fraction of value an employee creates for itself, the company will capture no value. Minimum wage also tells the prospective employee that society is not OK with him being a low value add loser at high likelihood of being a public burden, and that he should improve his productivity through education and training before entering the labor market. And it tells the company that to be profitable, it needs to invest in training or information technology to improve worker productivity, instead of simply looking for a cheaper worker.