Rather than change the meaning of \thesection, I leave it as is, without the dot, and change the manner that the section heading is output, by specifying a suffix to be appended to \thesection, for example.
\documentclass[11pt]{article}
%\renewcommand\thesection{\normalsize{\arabic{section}.}}
\makeatletter
%% SECTIONING \hskip'S BEFORE SECTIONING TITLES
\def\sectionkern {1em}
\def\subsectionkern {1em}
\def\subsubsectionkern{1em}
\def\paragraphkern {1em}
%% SUFFIX ADDED TO \the SECTIONING MACROS
\def\sectionsuffix {.}
\def\subsectionsuffix {}
\def\subsubsectionsuffix{}
\def\paragraphsuffix {}
%% SECTIONING LABEL: \the[section]\[section]suffix\hskip\[section]kern
\renewcommand\@seccntformat[1]{\normalsize\csname the#1\endcsname%
\csname#1suffix\endcsname\hskip\csname#1kern\endcsname\relax}
\makeatother
\begin{document}
\section{Introduction} \label{s-1}
I want to print just a number of Section \ref{s-1} without dot.
\end{document}

\thesectionto have the dot is that the default\thesubsectionwould then have two dots: "Subsection 1..1". – Teepeemm Oct 15 '20 at 13:53