Whoever replaces Jackson as the somewhat populist leader of the westerners and the common folk in general likely still becomes president, although possibly later on.
Will this replacement have the same authoritarian streak? If not, no Whig Party emerges, at least not under that name and with that central idea. (It was founded as a pro-Congress, anti-President party, and even the name mimicked the pro-Parliament British Whig Party.) That might lead Henry Clay to be somewhat less prominent.
Will his replacement be as opposed to internal improvements and the national bank? If not, he can effectively get Clay on his side by suggesting a compromise: more internal improvements in the south and west, funded via the national bank. This would prevent a lot of economic trouble between north and south, and would completely prevent the Whigs. (It might even happen in time to get Calhoun on board, since he was a nationalist before he became a radical states' rights advocate. Just imagine it: 'Clay and Calhoun, best friends forever.')