Will a 2003 Silverado tailgate fit 2001 Silverado?

No, a 2003 Silverado tailgate will likely not fit a 2001 Silverado. While they're both Silverados, GM made changes to body styles between the 2001 and 2003 model years. The tailgate's mounting points and overall dimensions are different enough that it's highly unlikely a direct swap will work without significant modifications (which are usually not practical).