No, but the long term character of the Great Depression would be quite different, beyond the United States itself.
While Smoot-Hawley was far from good policy, and set off a destructive round of economic isolationism worldwide, the Great Depression had more grounding in monetary factors. Given that no S-H does not significantly alter the nature of national monetary regimes and their particular implementation of the Gold Standard, nor change the enormously risky debt structure that emerged out of WWI, there will still be a quite nasty shock that generally drives the world into massive contraction. Said contraction likely still pushes the abandonment of the gold standard, and plays great havoc with the necessary role of financial intermediation.
After what amounts to a catastrophic financial crisis and banking panic, prospects for recovery in the '30s would be somewhat better absent the trade environment engendered by S-H, provided tariffs aren't simply boosted later on.
An interesting question is if British Imperial Preference would be at all the same if international trade were less encumbered.