No, this has been completely disproven. I hate his policies, but Peter Temin (The Jacksonian Economy) has shown that international silver flows---which dried up in the early 1830s and then the effects circled the globe to raise British interest rages---cause the depression. Virtually all economic historians, including Libertarians like Richard Timberlake, pretty much give Jackson a pass.
I'm not sure why the fevered speculation in State banks after Jackson dissolved the Central Bank wouldn't have played a factor.
A case could be made that depression could very well have been avoided or at least mitigated had Jackson not dissolved the Bank in favor of his "agricultural republic".