The closing of the Second Bank of the United States caused a major "bubble" in the market for land in the US. It also helped to cause a major problem with inflation.
As Jackson was killing the bank, he moved federal money into the "pet banks." As this happened these banks and others started printing more and more money of their own (this was legal at the time) and lending it to people who were speculating in land. Sales of public land went up tremendously. At the same time, states were borrowing money to build canals and roads because the Erie Canal had been such a big success. This bubble and all the printing of money led to inflation problems.
When Jackson then felt the need to issue the Specie Circular in 1837, the bubble burst and a panic ensued. This was the Panic of 1837. It is not really possible to say that the closing of the Second Bank caused the Panic of 1837, but it clearly caused the bubble and the inflationary pressures that helped lead to the panic.