Typically references to listings are done either to the listing as a whole (and therefore would have a caption) or to the line numbers within the listing.
Below I follow the second approach as it seems strange to provide a single-line listing with a caption. Even though the numbering is set on the left, you can also place it on the right. See section 5.6 Line numbers of the listings documentation for more options regarding the formatting of line numbers inside a listing.

\documentclass{article}
\usepackage[margin=1in]{geometry}% Just for this example
\usepackage{listings}
\lstnewenvironment{rcode}[1][]
{\lstset{
language=R,
numbers=left,
numberstyle=\footnotesize,
escapeinside=@@,
#1}}
{}
\newsavebox{\codebox}
\begin{document}
\begin{rcode}
lm.model <- lm(Total.Score ~ Game.Num, data = sid1133) @\label{r:lm.model-1}@
\end{rcode}
See Line~\ref{r:lm.model-1} or Line~\ref{r:lm.model-2}.
\begin{rcode}[firstnumber=200]
lm.model <- lm(Total.Score ~ Game.Num, data = sid1133) @\label{r:lm.model-2}@
\end{rcode}
\begin{lrbox}{\codebox}
\begin{lstlisting}[language=R]
lm.model <- lm(Total.Score ~ Game.Num, data = sid1133)
\end{lstlisting}
\end{lrbox}
Alternatively, see~(\ref{eq:lm.model-1}):
\begin{equation}
\usebox{\codebox} \label{eq:lm.model-1}
\end{equation}
\begin{lrbox}{\codebox}
\lstinline[language=R]|lm.model <- lm(Total.Score ~ Game.Num, data = sid1133)|
\end{lrbox}
Alternatively, see~(\ref{eq:lm.model-2}):
\begin{equation}
\usebox{\codebox} \label{eq:lm.model-2}
\end{equation}
\end{document}
In order to properly reference the line numbers, you'll have to escape to insert an appropriate \label.
If you really wish to include the listing code as an equation, you may want to consider storing the listing in a box before setting it (above I use \begin{lrbox}{\codebox} ... \end{lrbox}). Note the difference between the way content is set under the lstlisting environment and \lstinline macro.