Define a suitable theorem style that uses the default for thmtools, instead of the default from amsthm.
\documentclass[11pt,a4paper]{report}
\usepackage[
a4paper,
bindingoffset=0.2in,
left=1in,
right=1in,
top=1in,
bottom=1in,
footskip=.25in,
heightrounded, % <--- don't forget this
]{geometry}
\usepackage{amsmath,amssymb,amsthm,titling,url,array,xspace}
\usepackage{thmtools}
\declaretheoremstyle[
headfont=\bfseries,
bodyfont=\itshape,
]{myplain}
\declaretheorem[
style=myplain,
within=section,
name=Theorem,
]{theorem}
\declaretheoremstyle[
headfont=\bfseries,
bodyfont=\upshape,
headformat=(\NAME\ \NUMBER),
headpunct=,
]{example}
\declaretheorem[
style=example,
name=Example,
]{example}
\renewcommand{\theexample}{\roman{example}}
\begin{document}
\setcounter{chapter}{2} \setcounter{section}{2}
Some unspecified text that should occupy at least one line and some more
ending with a \textbf{decomposition} of $G$ if $H\oplus K=G$.
\begin{example}
$\mathbb{Z}$ is not decomposable, because, for all positive integers $m$ and $n$,
$\{0\}\ne mn\mathbb{Z}\subseteq m\mathbb{Z}\cap n\mathbb{Z}$.
\end{example}
$G$ is cyclic with $|G|=p^\alpha$ implies $G$ is something we won't tell.
\begin{theorem}[Primary Decomposition]
Something fun happens when we consider all primes.
\end{theorem}
\end{document}

Some notes: \bigoplus is the wrong symbol for the direct sum of two subgroups.
The package epsfig has been deprecated for more than 20 years and should not be used in new documents. Use graphicx instead, if you need to insert graphics.
The package epstopdf doesn't need to be loaded if you already load graphicx. Neither amsfonts if you load amssymb.
Never use \\ for vertical spacing or for ending a paragraph.
\\at the end of a paragraph, it creates spurious lines that are "infinitely bad" (badness 10000) and warnings in the log. – David Carlisle Sep 28 '17 at 20:04\paror\newlineso could you kindly let me know how to reduce the spacing height to the blue one? – James Sep 28 '17 at 20:20\newlineor\parjust use a blank line (which will do the same as\par) it is impossible to tell you what to change given the fragments posted (questions should always have a complete test document that shows the issue) by default you get no vertical space at a paragraph break so if you are getting space then your document is setting it, and you need to set it to a smaller value – David Carlisle Sep 28 '17 at 20:23