Would slowing down immigration and blocking illegals from working in the US automatically cause the minimum wage to rise on its own?
Minimum wages are wage floors set by law, not by the market. They, however, are subject to political pressure from citizens. If, say, hypothetically, all immigration, legal or otherwise, was stopped, then there would be a stronger demand to hire natives to fill the positions currently taken up by immigrants. Esp. the low-skilled, low-wage jobs such as in construction, food processing, hospitality. Natives have made it clear they are not willing to work at the wages that immigrants currently do. So, the wages would have to rise to entice natives to sign up to work. The market will take care of that. Over time, minimum wages will rise to reflect the higher market wages.