You need to redefine the page style plain
used on the chapter title page like:
\fancypagestyle{plain}{% <==============================================
\fancyhf{}
\fancyfoot[R]{\thepage}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}%
}
With the complete code
\documentclass[a4paper,12pt]{report}
\usepackage[%
left=1.5in,right=1in,top=1in,bottom=1in,
includefoot,heightrounded
]{geometry}
\usepackage{lipsum}
\usepackage{fancyhdr}
\renewcommand{\baselinestretch}{1.3}
\usepackage{titlesec}
\titleformat{\chapter}[display]{\normalfont\huge\bfseries\centering}{\chaptertitlename\ \thechapter}{20pt}{\Huge}
\titlespacing{\chapter}{0pt}{0pt}{1cm}
\fancypagestyle{plain}{% <==============================================
\fancyhf{}
\fancyfoot[R]{\thepage}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}%
}
\begin{document}
\pagenumbering{roman}
\pagestyle{fancy}
\fancyfoot{}
\fancyfoot[R]{\thepage}
\fancyhead[L]{\textit{\chaptername\thechapter}}
\chapter{Introduction}
\pagenumbering{arabic}
\setcounter{page}{1}
\lipsum[2-4]
\lipsum[2-4]
\end{document}
you get:

Update:
To add an chapter/abstract before chapter 1 please change your used code to:
%\begin{center} % <======================================================
\chapter*{Abstract} % <============================= usualy not numbered!
%\end{center}
%\addcontentsline{toc}{section}{Abstract}
\lipsum[2-4] % <========================================================
With the complete code
\documentclass[a4paper,12pt]{report}
\usepackage[%
left=1.5in,right=1in,top=1in,bottom=1in,
includefoot,heightrounded
]{geometry}
\usepackage{lipsum}
\usepackage{fancyhdr}
\renewcommand{\baselinestretch}{1.3}
\usepackage{titlesec}
\titleformat{\chapter}[display]{\normalfont\huge\bfseries\centering}{\chaptertitlename\ \thechapter}{20pt}{\Huge}
\titlespacing{\chapter}{0pt}{0pt}{1cm}
\fancypagestyle{plain}{% <==============================================
\fancyhf{}
\fancyfoot[R]{\thepage}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}%
}
\begin{document}
\pagenumbering{roman}
\pagestyle{fancy}
\fancyfoot{}
\fancyfoot[R]{\thepage}
\fancyhead[L]{\textit{\chaptername\thechapter}}
%\begin{center} % <======================================================
\chapter*{Abstract}
%\end{center}
%\addcontentsline{toc}{section}{Abstract}
\lipsum[2-4] % <========================================================
\chapter{Introduction}
\pagenumbering{arabic}
\setcounter{page}{1}
\lipsum[2-4]
\lipsum[2-4]
\end{document}
you get then:

Chapter 1
on both sides of the header?\thispagestyle{fancy}
after of\chapter{...}
Chapter1
on the left andCHAPTER 1 INTRODUCTiON
on th right