No.
But what you can do it make the header height 1 pixel. Remove all images and text from the header. Remove the search site box from the header. Do not elect to display the page name at the top of each page. Go to edit page/layout. Counting from the top down, select layout 4, 5, 6, 7, 8, or 9. In the uppermost horizontal content area create a "Header" in English for each English page, and create a "Header" in French for each French page.
If you select a theme, or select colors yourself, where the area around the edges of the page are not starkly contrasting in color with the content area it may look OK. You'll really have to play around with colors and fonts and themes to make it look half-way decent.
Have you considered making two sites? One could be
sites.google.com/exampleenglish and the other
sites.google.com/examplefrench. Then, using links, you could more-or-less integrate the two so that they appear to be a single site. This would give you the ability to set up a unique header for each (you could do a lot of unique things for each).