Tikzcd pullback square issue

Your approach nests tikzpictures. The standard way to avoid that is to use \saveboxes.

\documentclass{article}
\usepackage{tikz-cd}
\newsavebox{\pullback}
\sbox\pullback{%
\begin{tikzpicture}%
\draw (0,0) -- (1ex,0ex);%
\draw (1ex,0ex) -- (1ex,1ex);%
\end{tikzpicture}}
\begin{document}
\begin{center}
\begin{tikzcd}
F \arrow[r] \arrow[d] 
\arrow[dr, phantom, "\usebox\pullback" , very near start, color=black]
& * \arrow[d] \\
X \arrow[r, "f"] & Y \\
\end{tikzcd}
\end{center}
\end{document}

You defined your command with a parameter, which you did not use. Let's assume that there is a need for such parameters. Then \saveboxes are unhandy. Therefore I add an alternative: a path picture with as many things fixed as you can think of. E.g. - is to say that there should not be an arrow, etc. (I understand that it is very unlikely that you want the symbol to be red, this is just to illustrate that parameters work here.)

\documentclass{article}
\usepackage{tikz-cd}
\tikzset{pullback/.style={minimum size=1.2ex,path picture={
\draw[opacity=1,black,-,#1] (-0.5ex,-0.5ex) -- (0.5ex,-0.5ex) -- (0.5ex,0.5ex);%
}}}
\begin{document}
\begin{center}
\begin{tikzcd}
F \arrow[r] \arrow[d] 
\arrow[dr, phantom," " {pullback=red}, very near start, color=black]
& * \arrow[d] \\
X \arrow[r, "f"] & Y \\
\end{tikzcd}
\end{center}
\end{document}

Why don't you simply insert the \lrcorner symbol?

 \documentclass[svgnames]{article}
\usepackage{tikz-cd, amsmath, amssymb}%

\begin{document}

\begin{center}
\begin{tikzcd}
F \arrow[r] \arrow[d]
\arrow[dr, phantom, "\scalebox{1.5}{\color{IndianRed}$\lrcorner$}" , very near start, color=black]
& * \arrow[d] \\
X \arrow[r, "f"] & Y \\
\end{tikzcd}
\end{center}

\end{document} 

Inside a tikzpicture, along a path, you can put not only node but also pic.

\documentclass[tikz,border=5mm]{standalone}
\begin{document}
\begin{tikzpicture}[bmr/.pic={
\draw (2mm,0)--++(-90:2mm)--++(180:2mm);
}]
\path
(0,0)     node (F) {$F$}
+(-45:.2) pic[scale=.8,red]{bmr}
+(0:1.5)  node (star) {$*$}
++(-90:1) node (X) {$X$}
+(0:1.5)  node (Y) {$Y$};
\draw[->] (F)--(star);
\draw[->] (F)--(X);
\draw[->] (X)--(Y) node[midway,above,scale=.6]{$f$};
\draw[->] (star)--(Y);
\end{tikzpicture}   
\end{document}