Actually you do not need a minipage here.
\documentclass[12pt]{article}
\usepackage[
top=0.75in, bottom=0.75in, left=0.75in, right=0.75in,
textheight=8.75in
]{geometry}
\setlength{\parindent}{0pt}
\setlength{\parskip}{0pt}
\pagestyle{empty}
\begin{document}
{\LARGE\textsc{My Name}}
\hfill
\footnotesize\texttt{myemail@abc.edu}
\smallskip
\hrule
\bigskip
{\textsc{ABC University}}
\end{document}

In case you want to retain your minipages, you should make their width almost half of the textwidth as \begin{minipage}[b]{0.495\linewidth}.
\documentclass[12pt]{article}
\usepackage[
top=0.75in, bottom=0.75in, left=0.75in, right=0.75in,
textheight=8.75in
]{geometry}
\setlength{\parindent}{0pt}
\setlength{\parskip}{0pt}
\pagestyle{empty}
\begin{document}
\begin{minipage}{0.495\linewidth}
{\LARGE\textsc{My Name}}
\end{minipage}
\begin{minipage}[b]{0.495\linewidth}
\begin{flushright}
\footnotesize\texttt{myemail@abc.edu}
\end{flushright}
\end{minipage}%
\smallskip
\hrule
\bigskip
{\textsc{ABC University}}
\end{document}

Or you can issue a \hfill inbetween the minipages like
\begin{minipage}{0.4\linewidth}
{\LARGE\textsc{My Name}}
\end{minipage}\hfill % ,----here
\begin{minipage}[b]{0.4\linewidth}
\begin{flushright}
\footnotesize\texttt{myemail@abc.edu}
\end{flushright}
\end{minipage}%
\smallskip
\hrule
\bigskip
{\textsc{ABC University}}
to get the same effect as mentioned in the comment by @egreg.
\noindentright before the minipage? – Mico Apr 26 '12 at 10:51