How to get autobreaking with sequential tables

\raggedright%%%%%%%%%%%%%%%%%%

There is no difference whatsoever between \hspace{1em} and \hskip1em (so long as the latter is not followed by plus or minus: with the former you get the internal version

\@hspace{1em}

which finally expands to \hskip 1em\relax.

What's the problem in your document? A tabular environment is, to TeX, just like a big letter. It wants simply to typeset paragraphs following its rules. Indeed you get two messages:

Overfull \hbox (168.00093pt too wide) in paragraph at lines 24--30
[] [] [][]

Overfull \hbox (26.00035pt too wide) in paragraph at lines 32--40
[] [] [] [] [] [][]

that simply mean: "Dear Peter, I couldn't typeset your paragraphs without violating the current \tolerance, so I've left lines sticking out in the right margin."

Exactly the same happens if you define

\newcommand*{\Space}{\hspace{1.0em}}

because you're just telling TeX the same thing.

Justification is obtained easily by allowing stretchability to \Space:

\documentclass{article}
\usepackage{showframe}

\newcommand*{\Space}{\hspace{1.0em plus 3em}}
\newcommand*{\MyTabularA}{%
    \begin{tabular}{ccc}
        Column 1 & Column 2 & Column 3 \\
        1 & 2 & A\\
        3 & 4 & B\\
        5 & 6 & C\\
    \end{tabular}%
}

\newcommand*{\MyTabularB}{%
    \begin{tabular}{ccc}
        1 & 2 & 3 \\
        1 & 2 & A\\
        3 & 4 & B\\
        5 & 6 & C\\
    \end{tabular}%
}

\begin{document}

\noindent
\MyTabularA\Space
\MyTabularA\Space
\color{red}\MyTabularA\Space
\color{blue}\MyTabularA\Space
\MyTabularA

\bigskip

\color{black}\noindent
\MyTabularB\Space
\MyTabularB\Space
\MyTabularB\Space
\MyTabularB\Space
\MyTabularB\Space
\color{red}\MyTabularB\Space
\color{blue}\MyTabularB

\end{document}

(I've removed the useless things, as usual. ;-))

Of course, the final line in the second paragraph doesn't stretch the space, because of TeX's rules.