You can update the verbatim font to use a different \linespread. More specifically, include \linespread{1} as part of \verbatim@font:

\documentclass{uwthesis}% http://ctan.org/pkg/uwthesis
\makeatletter
\def\verbatim@font{\linespread{1}\normalfont\ttfamily}
\makeatother
\begin{document}
Double \
Spaced \
Lines
\begin{verbatim}
These
lines
are
not
doublespaced
\end{verbatim}
Double \
Spaced \
Lines
\end{document}
The uwthesis class relies on the default LaTeX kernel verbatim environment, hence the ease of not having to add extra packages to obtain a desired output. For a reference to \linespread, see Why doesn’t \linespread work? on the TeX FAQ.
You could contain the redefinition in a macro that would act as a switch:
\makeatletter
\newcommand{\nextverbatimspread}[1]{%
\def\verbatim@font{%
\linespread{#1}\normalfont\ttfamily% Updated definition
\gdef\verbatim@font{\normalfont\ttfamily}}% Revert to old definition
}
\makeatother
which you could use in the form
\nextverbatimspread{1}
\begin{verbatim}
% ...single-spaced verbatim
\end{verbatim}
% ...some other content
\begin{verbatim}
% ...double-spaced verbatim
\end{verbatim}
For creating your own verbatim-like environment is a little more tricky. The easiest would be to include the verbatim package and use David's suggestion of wrapping the verbatim environment in command-form:

\documentclass{uwthesis}% http://ctan.org/pkg/uwthesis
\usepackage{verbatim}% http://ctan.org/pkg/verbatim
\makeatletter
\newcommand{\nextverbatimspread}[1]{%
\def\verbatim@font{%
\linespread{#1}\normalfont\ttfamily% Updated definition
\gdef\verbatim@font{\normalfont\ttfamily}}% Revert to old definition
}
\newenvironment{myverbatim}[1][1.5]
{\def\verbatim@font{%
\linespread{#1}\normalfont\ttfamily% Updated definition
\gdef\verbatim@font{\normalfont\ttfamily}}% Revert to old definition
\verbatim}
{\endverbatim}
\makeatother
\begin{document}
Double \
Spaced \
Lines
\nextverbatimspread{1}
\begin{verbatim}
These
lines
are
not
doublespaced
\end{verbatim}
Double \
Spaced \
Lines
\begin{myverbatim}
These
lines
are
doublespaced
\end{myverbatim}
Double \
Spaced \
Lines
\begin{myverbatim}[1]
These
lines
are
not
doublespaced
\end{myverbatim}
\end{document}
The myverbatim environment takes an option argument to specify a modified \linespread. The default is 1.5, as specified by uwthesis ("not quite doublespaced", according to uwthesis.cls, line 158).
\fvset{baselinestretch=1}in the preamble, thus avoiding the need to type it every time. – egreg Dec 12 '12 at 09:04