Scaling had to be done also for nodes shapes (by transform shapes):
\documentclass[border=3.141502, varwidth]{standalone}
\usepackage{tikz}
\usetikzlibrary{graphs,positioning}
\begin{document}
\begin{tikzpicture}[scale=0.75, transform shape,%x=7.5mm,
every node/.style = {circle, draw, very thick,
inner sep = 0, outer sep=0pt,
minimum size = 10mm},
]
\graph[grow right sep=10mm]{
1 -> 2 -> 3 -> 4
};
\draw[help lines] (-0.5,-1.5) grid (8,1.5);
\end{tikzpicture}
\medskip
\begin{tikzpicture}[
every node/.style = {circle, draw, very thick,
inner sep = 0, outer sep=0pt,
minimum size = 10mm},
]
\graph[grow right sep=10mm]{
1 -> 2 -> 3 -> 4
};
\draw[help lines] (-0.5,-1.5) grid (8,1.5);
\end{tikzpicture}
\end{document}
Addendum:
Unfortunately the question wasn't entirely clear (at least me). Now I guess, that you like to have smaller shapes, smaller node distances but normal text font size. This can be achieved by accordingly changes size of nodes, size of node distance and (if you like) also step of help lines. all this changes are in code marked by % <---:
\documentclass[border=3.141502, varwidth]{standalone}
\usepackage{tikz}
\usetikzlibrary{graphs,positioning}
\begin{document}
\begin{tikzpicture}[
every node/.style = {circle, draw, very thick,
inner sep = 0, outer sep=0pt,
minimum size = 7.5mm}, % <---
]
\graph[grow right sep=7.5mm]{ % <---
1 -> 2 -> 3 -> 4
};
\draw[help lines, step=7.5mm] % <---
(-0.5,-1.5) grid (6,1.5);
\end{tikzpicture}
for comparison with "scaled" image:
\begin{tikzpicture}[
every node/.style = {circle, draw, very thick,
inner sep = 0, outer sep=0pt,
minimum size = 10mm},
]
\graph[grow right sep=10mm]{
1 -> 2 -> 3 -> 4
};
\draw[help lines] (-0.5,-1.5) grid (8,1.5);
% for test of scaling:
\end{tikzpicture}
\end{document}
Is this result what you after? As you see, it cannot be get by simple scaling. However you still can use first solution and in nodes use bigger font size, for example Large in your case.
