The answer is no. Basically, not every part of the Osgood curve is an Osgood curve itself, because of the joins. The first example of the Osgood curve with the property that its every sub-arc is also an Osgood curve was provided by Sierpinski. His construction was 
improved by Knopp. 

*Space-filling curves* by Hans Sagan contains a good survey of the results.