Table with header row of normal height, remaining rows fill page vertically, table fills textwidth

Question

Is there a way to achieve a table of width \textwidth, with the top row being of normal height, and the remaining rows filling the rest of the page in equal proportion?

I have this implemented but with an inexact fudge factor. I can imagine some implementation using a fancy tabular package (I was not able to make tabularht work) or something like the below where \rowht is set precisely based on \arraystretch or something and the height of the header, but I don't know how to do that.

\documentclass{article}
\usepackage{tabularx}
\usepackage[margin=1in]{geometry}
\def\rowht{0.935in}
\pagestyle{empty}
\begin{document}
\noindent
\begin{tabularx}{\textwidth}{| X | X | X | X |}
  \hline
  A & B & C & D \\\hline
  & & &\\[\rowht]\hline
  & & &\\[\rowht]\hline
  & & &\\[\rowht]\hline
  & & &\\[\rowht]\hline
  & & &\\[\rowht]\hline
  & & &\\[\rowht]\hline
  & & &\\[\rowht]\hline
  & & &\\[\rowht]\hline
\end{tabularx}
\end{document}

Answer 1

I will give a solution with the new LaTeX3 package tabularray:

\documentclass{article}
\usepackage[margin=1in]{geometry}
%\usepackage{showframe}
%\renewcommand*{\ShowFrameColor}{\color{red}}
\pagestyle{empty}
\usepackage{tabularray}
\def\rowht{%
  \dimexpr
    (
      \textheight
       - \baselineskip                   % height of first row
       - 4pt * \value{rowcount}          % rowsep sizes
       - 0.4pt * ( \value{rowcount} - 1) % hrule sizes
    ) / ( \value{rowcount} - 1 )
  \relax
}
\begin{document}

\noindent
\begin{tblr}{
  colspec={|X|X|X|X|},
  hlines,
  row{2-Z} = {\rowht}, % Z stands for the last row
}
  A & B & C & D \\
    &   &   &   \\
    &   &   &   \\
    &   &   &   \\
    &   &   &   \\
    &   &   &   \\
    &   &   &   \\
    &   &   &   \\
    &   &   &   \\
\end{tblr}

\newpage

\noindent
\begin{tblr}{
  colspec={|X|X|X|X|},
  hlines,
  row{2-Z} = {\rowht},
}
  A & B & C & D \\
    &   &   &   \\
    &   &   &   \\
    &   &   &   \\
    &   &   &   \\
    &   &   &   \\
\end{tblr}

\end{document}