The following provides \stretchtofitimage which stretches the image vertically to fit on the "remainder of the page" by setting the height key-value using some calculations:

\documentclass{article}
\usepackage{geometry,graphicx,zref-savepos}% http://ctan.org/pkg/{geometry,graphicx,zref}
\makeatletter
% \zsaveposx is defined since 2011/12/05 v2.23 of zref-savepos
\@ifundefined{zsaveposy}{\let\zsaveposy\zsavepos}{}
\newcounter{stretchtofit}
\newcommand{\stretchtofitimage}[2][]{%
\stepcounter{stretchtofit}% Next image
\zsaveposy{fig:top\thestretchtofit}% Save vertical position on page
\includegraphics[height=\dimexpr\zposy{fig:top\thestretchtofit}sp-\Gm@bmargin-1pt,#1]{#2}}
\begin{document}
\section{A section}
\subsection{A subsection}
Here is some text.
\subsubsection{A subsubsection}
{\centering\stretchtofitimage[width=.6\linewidth]{example-image-a}\par}
Here is some more text.
\end{document}
The calculation of the height uses zref's savepos module to store the vertical height (in scaled points) from the bottom of the page. Since this vertical height is measured from the bottom up, we subtract the bottom margin (provided by geometry's \Gm@bmargin) as well as a 1pt smidgen. There is some stretch allowed below sections, which this 1pt removal accommodates for.
Note that the above assumes you are not using a \caption - it wasn't included in your code snippet, so this may be a fair assumption.
The image above includes a showframe, which is also supported by geometry.
If you want to have a single-page image with \subsubsection title, then you can get around without zref:

\documentclass{article}
\usepackage{graphicx}% http://ctan.org/pkg/graphicx
\begin{document}
\section{A section}
\subsection{A subsection}
Here is some text.
\subsubsection{A subsubsection\strut}
{\centering
\includegraphics[width=\textwidth,height=\dimexpr\textheight-1.7ex-\baselineskip-1pt]{example-image-a}\par}
Here is some more text.
\end{document}
The measurements for calculating the height is based on the fact that a \subsubsection heading is set using \normalfont\normalsize\bfseries in the standard document classes (which has a regular \baselineskip) and a post-title skip of 1.5ex \@plus .2ex. Again, the 1pt difference is for good measure.
The \strut inside \subsubsection ensures proper usage of the \baselineskip for the title. You may not need it in your actual title.