Use evaluate option of \foreach to display integers not floats in TikZ

To get integer results, use the TikZ function int(...) when computing \r for the node text:

\documentclass{article}
\usepackage{tikz}
\begin{document}

 \begin{tikzpicture}
      \draw[->] (-2,0) -- (2,0) node[right] {$x$};  
      \draw[->] (0,-2) -- (0,2) node[above] {$y$};
      \foreach \x in {-3,...,3} 
               \draw (\x/2,1pt) -- (\x/2,-2pt);
      \foreach \y in {-3,...,3} 
               \draw (1pt,\y/2) -- (-2pt,\y/2);
      \draw[thick,blue] (0,0) circle (1.5cm);
      \foreach \x [evaluate=\x as \r using int(\x/1.5)] in {-1.5, 1.5} 
               \draw (\x cm,1pt) -- (\x cm,-1pt) 
                      node[anchor=north, fill=white] {\tiny $\r$};

\end{tikzpicture}%
\end{document} 

Just for completeness. In this case there is no need to evaluate anything since coordinates get parsed automatically, so you could simply do

\foreach \X in {-1, 1} 
               \draw (\X*1.5cm,1pt) -- (\X*1.5cm,-1pt) 
                      node[anchor=north, fill=white,font=\tiny] {$\X$};

Please note also that \x may not be a good loop variable if you ever intend also to make use of calc.

\documentclass{article}
\usepackage{tikz}
\begin{document}

 \begin{tikzpicture}
      \draw[->] (-2,0) -- (2,0) node[right] {$x$};  
      \draw[->] (0,-2) -- (0,2) node[above] {$y$};
      \foreach \X in {-3,...,3} 
               \draw (\X/2,1pt) -- (\X/2,-2pt);
      \foreach \y in {-3,...,3} 
               \draw (1pt,\y/2) -- (-2pt,\y/2);
      \draw[thick,blue!80!black] (0,0) circle (1.5cm);
      \foreach \X in {-1, 1} 
               \draw (\X*1.5cm,1pt) -- (\X*1.5cm,-1pt) 
                      node[anchor=north, fill=white,font=\tiny] {$\X$};

\end{tikzpicture}%
\end{document}