Scaling tables between page sizes, and filling the page

Question

I'm a teacher, and this is a design pattern I hope to use a lot. I have some information that I want to present as 1) a poster (a1), and 2) a handout (a4). What I would like to be able to do is set just the paper size, and get an appropriate scaling of my font. The data I'm presenting, is also almost always going to be laid out using some sort of grid (tabular) layout. So for example. I'm working on a poster with all of the greek letters. Here is what I have so far:

and this is the code I used to generate it:

\documentclass[extrafontsizes,60pt]{memoir}
\usepackage{amsmath}
\usepackage{lmodern}
\usepackage[T1]{fontenc}
\usepackage{geometry}
\geometry{a1paper, portrait, margin=0.5in}
\begin{document}
\Huge
\centering
Greek Letters
\begin{center}
  \begin{tabular}{ c c c c }
    $A$ $\alpha$ & $B$ $\beta$ & $\Gamma$ $\gamma$ & $\Delta$ $\delta$ \
    \large alpha & \large beta & \large gamma & \large delta  \
    $E$ $\epsilon$ & $Z$ $\zeta$ & $H$ $\eta$ & $\Theta$ $\theta$ \
    \large epsilon & \large zeta & \large eta & \large theta  \
    $A$ $\alpha$ & $B$ $\beta$ & $\Gamma$ $\gamma$ & $\Delta$ $\delta$ \
    \large alpha & \large beta & \large gamma & \large delta  \
    $E$ $\epsilon$ & $Z$ $\zeta$ & $H$ $\eta$ & $\Theta$ $\theta$ \
    \large epsilon & \large zeta & \large eta & \large theta  \
    $A$ $\alpha$ & $B$ $\beta$ & $\Gamma$ $\gamma$ & $\Delta$ $\delta$ \
    \large alpha & \large beta & \large gamma & \large delta  \
    $E$ $\epsilon$ & $Z$ $\zeta$ & $H$ $\eta$ & $\Theta$ $\theta$ \
    \large epsilon & \large zeta & \large eta & \large theta  \
  \end{tabular}
\end{center}
\end{document}

The things I'm not sure how to do are: 1) the table doesn't fill the width of the page, 2) the "label row" under each "character row" should be tight to the row above, not the row below, and 3) if I change from a1paper to a4paper, I want the font to scale appropriately so that everything stays in proportion. Any help would be appreciated.

EDIT: here is the current code using John's suggestion The table row spacing/vertical alignment is sorta the last issue

\documentclass[extrafontsizes,12pt]{memoir}
\usepackage{amsmath}
\usepackage{lmodern}
\usepackage[T1]{fontenc}
\usepackage{adjustbox}
\usepackage{geometry}
\geometry{a1paper, portrait, margin=0.5in}
\pagestyle{empty}
\begin{document}
\centering
\begin{center}
  \adjustbox{min width=\textwidth}{
  \def\arraystretch{0.5}
  \begin{tabular}{ c c c c }
    \multicolumn{4}{c}{\huge Greek Letters} \
    \hline \\
    \Large $A$ $\alpha$ & \Large $B$ $\beta$ & \Large $\Gamma$ $\gamma$ & \Large $\Delta$ $\delta$ \
    \tiny alpha & \tiny beta & \tiny gamma & \tiny delta  \\\
    \Large $E$ $\epsilon$ & \Large $Z$ $\zeta$ & \Large $H$ $\eta$ & \Large $\Theta$ $\theta$ \
    \tiny epsilon & \tiny zeta & \tiny eta & \tiny theta  \\\
    \Large $I$ $\iota$ & \Large $K$ $\kappa$ & \Large $\Lambda$ $\lambda$ & \Large $M$ $\mu$ \
    \tiny iota & \tiny kappa & \tiny lambda & \tiny mu  \\\
    \Large $N$ $\nu$ & \Large $\Xi$ $\xi$ & \Large $O$ $o$ & \Large $\Pi$ $\pi$ \
    \tiny nu & \tiny xi & \tiny omicron & \tiny pi  \\\
    \Large $P$ $\rho$ & \Large $\Sigma$ $\sigma$ & \Large $T$ $\tau$ & \Large $\Upsilon$ $\upsilon$ \
    \tiny rho & \tiny sigma & \tiny tau & \tiny upsilon  \\\
    \Large $\Phi$ $\phi$ & \Large $X$ $\chi$ & \Large $\Psi$ $\psi$ & \Large $\Omega$ $\omega$ \
    \tiny phi & \tiny chi & \tiny psi & \tiny omega
  \end{tabular}
  }
\end{center}
\end{document}