The idea is to create a bunch of special column types such that every time this column type appears in a tabular it will always have the same width. This requires 2 runs of the document, where the widths computed in the first run are saved in the aux file.
Note: If you ever want to make the columns narrower, you will need to delete the aux file or at least comment out the \AtEndDocument.
\documentclass{article}
\usepackage{array}
\newsavebox{\tempbox}% \box0 etc. used
\makeatletter
\newcommand{\saveWidth}[1]% #1=column name (A,B,...)
{\immediate\write\@auxout{\string\initWidth{#1}{\the\csname Width#1\endcsname}}}
\newcommand{\initWidth}[2]% #1=column name (A,B,...), #2=the width
{\@ifundefined{Width#1}{}{\global\csname Width#1\endcsname=#2\relax}}
\makeatother
\newlength{\WidthA}% one for each column type
\newlength{\WidthB}
\newlength{\WidthC}
\AtEndDocument{\saveWidth{A}\saveWidth{B}\saveWidth{C}}
\newcolumntype{A}{>{\savebox{\tempbox}\bgroup}{l}<{\egroup%
\ifdim\wd\tempbox>\WidthA \global\WidthA=\wd\tempbox\fi%
\makebox[\WidthA][l]{\usebox\tempbox}}}
\newcolumntype{B}{>{\savebox{\tempbox}\bgroup}{c}<{\egroup%
\ifdim\wd\tempbox>\WidthB \global\WidthB=\wd\tempbox\fi%
\makebox[\WidthB][c]{\usebox\tempbox}}}
\newcolumntype{C}{>{\savebox{\tempbox}\bgroup}{r}<{\egroup%
\ifdim\wd\tempbox>\WidthC \global\WidthC=\wd\tempbox\fi%
\makebox[\WidthC][r]{\usebox\tempbox}}}
\begin{document}
\noindent\begin{tabular}{ABC}
left & center & right\\
wide left & wide center & wide right
\end{tabular}
\medskip
Some text here.
\medskip
\noindent\begin{tabular}{ABC}
left & center & right\\
\end{tabular}
\medskip
Some text here.
\medskip
\noindent\begin{tabular}{ABC}
very wide left & very wide center & very wide right\\
\end{tabular}
\end{document}
To make it easier to create new columns, I added \newcolumnwidth{<name>}{l/c/r}
(which is itself rather ugly).
\newcommand{\newcolumnwidth}[2]% #1=new column type, #2=l/c/r
{\expandafter\newlength\csname Width#1\endcsname%
\AtEndDocument{\saveWidth{#1}}%
\newcolumntype{#1}{>{\savebox{\tempbox}\bgroup}{#2}<{\egroup%
\ifdim\wd\tempbox>\csname Width#1\endcsname \global\csname Width#1\endcsname=\wd\tempbox\fi%
\makebox[\csname Width#1\endcsname][#2]{\usebox\tempbox}}}}
\newcolumnwidth{A}{l}
\newcolumnwidth{B}{c}
\newcolumnwidth{C}{r}
1em
without increases from equations, multirow, or anything else? Either way: no. I'm looking for a robust solution which can handle anything.