Simple way to make circular arrow arc?

You can compute the intersections of the node boundaries and the circle to draw the arcs. For your convenience, I packed all the stuff in a pic. I kept the red circle in to prove that the arcs perfectly cover it.

\documentclass[border=5mm,tikz]{standalone}
\usetikzlibrary{arrows.meta,bending,calc,intersections}
\begin{document}
\begin{tikzpicture}[every node/.style={circle,draw,fill=white},
 pics/circular arc/.style args={from #1 to #2}{code={
 \path[name path=arc] 
  let \p1=(#1),\p2=(#2),\n1={atan2(\y1,\x1)},\n2={atan2(\y2,\x2)},
    \n3={ifthenelse(abs(\n1-\n2)<180,\n2,\n2-360)}
  in (\n1:\r) arc(\n1:\n3:\r);
 \draw[-{Stealth[bend]},pic actions,
    name intersections={of=#1 and arc,by=arcstart},
    name intersections={of=#2 and arc,by=arcend}] 
    let \p1=(arcstart),\p2=(arcend),\n1={atan2(\y1,\x1)},\n2={atan2(\y2,\x2)},
     \n3={ifthenelse(abs(\n1-\n2)<180,\n2,\n2-360)}
  in (\n1:\r) arc(\n1:\n3:\r);
 }}]
\def\r{3}
\draw[red] (0,0) circle (\r);

% 4 nodes with different sizes
\path
(180:\r) node[name path=Sp] (Sp) {Spring}
(90:\r)  node[name path=Su] (Su) {Summer}
(0:\r)   node[name path=Au] (Au) {Autumn}
(-90:\r) node[name path=Wi] (Wi) {Winter};

% How to make arrow arcs cicular? (the red one with arrow tips)
\begin{scope}[-stealth,bend left]
 \path pic{circular arc=from Sp to Su}
    pic{circular arc=from Su to Au}
    pic{circular arc=from Au to Wi}
    pic{circular arc=from Wi to Sp};

\end{scope}
\end{tikzpicture}
\end{document}

Like this?

\documentclass[border=5mm,tikz]{standalone}
\begin{document}
\begin{tikzpicture}
[every node/.style={circle,draw,fill=white}]
\def\r{3}
%\draw[red] (0,0) circle (\r);

% 4 nodes with different sizes
\path
(180:\r) node (Sp) {Spring}
(90:\r) node (Su) {Summer}
(0:\r) node (Au) {Autumn}
(-90:\r) node (Wi) {Winter};

\begin{scope}[-stealth,red,bend left]
\draw (Sp.90) to (Su.180);
\draw (Su.0) to (Au.90);
\draw (Au.270) to (Wi.0);
\draw (Wi.180) to (Sp.270);
\end{scope}
\end{tikzpicture}
\end{document}

I have just found a seem-to-be-simple way with angles library. Every circular arrow arc is an angle mark.

\documentclass[border=5mm,tikz]{standalone}
\usetikzlibrary{angles}
\begin{document}
\begin{tikzpicture}[every node/.style={circle,draw,fill=white}]
\def\r{3}
\draw[red] (0,0) circle (\r);

% 4 nodes with different sizes
\path
(0,0) coordinate (O)
(180:\r) node (Sp) {Spring}
(90:\r)  node (Su) {Summer}
(0:\r)   node (Au) {Autumn}
(-90:\r) node (Wi) {Winter};

% Make arrow arcs cicular using angles library as angle mark
\path
(Sp.north) coordinate (SpN)
(Su.west) coordinate (SuW)
pic[draw=blue,fill=none,stealth-,angle radius=\r cm]{angle=SuW--O--SpN}
(Su.east) coordinate (SuE)
(Au.north) coordinate (AuN)
pic[draw=blue,fill=none,stealth-,angle radius=\r cm]{angle=AuN--O--SuE};
\end{tikzpicture}
\end{document}