No it doesn’t: you don’t need to redo the lay out. But you’ll have to adjust the CSS from pixel widths to relative units: either using percentages for the widths and ems for padding, margin… or everything in ems.
The only somewhat safe way to lay out web pages so they work everywhere,
is to not lock sizes to anything but viewport - using percentage, and decide what is too wide or too narrow for your creation.
You have 4 basic methods:
1: Fluid layout: all widths in ‘%’. (One can also use auto-width in some
cases.)
- adapt well to various viewports, but can become overly wide and narrow.
2: Conditional fluid layout: all widths in ‘%’, + min-width/max-width
for entire layout in ‘em’ and/or ‘px’.
- adapt well to various viewports, and can not become overly wide or
narrow.
- if ‘em’ is used for max-width, it’s sometimes refered to as
“conditional elastic” since they behave as fluid and ‘em’-sized at the
same time.
3: Fixed layout: all widths in ‘px’.
4: Em-sized layout: all widths in ‘em’.
-can’t adapt to anything.
In fact you have more methods if you consider these variants:
5: Fixed - fluid - fixed: overall width in ‘%’, with fixed-width side
columns and fluid main column.
- adapts well to various viewports, but main column can become overly
wide and narrow.
- also comes in 2-columns fixed-fluid or fluid-fixed, and in other
column-mixes.
6: Conditional fixed - fluid - fixed: overall width in ‘%’, with
fixed-width side columns and fluid main column, + min-width/max-width
for entire layout in ‘em’ and/or ‘px’.
- adapt well to various viewports, and main column can not become
overly wide or narrow.
- also comes in 2-columns fixed-fluid or fluid-fixed, and in other
column-mixes.
7: Overly backwards and unnecessary complex “adaptive fixed/em-sized”
layout: width in ‘px’ or ‘em’, + max-width in ‘%’ and min-width in ‘em’
or ‘px’.
- can adapt to various viewports to a certain degree, but ‘em’ sized
tends to break under font resizing stress unless all font-sizes are left
at default.
So you have a lot to choose from lol.
IMHO a fixed layout should only be used if the client really, really insists on it.
It’s my believe that a site isn’t only build using valid HTML and CSS, but also needs to adhere to the accessibility and usability standards. You don’t have control on what kind of users will use the site 
But granted, fixed layouts are much more easier lol