Issue on Getting Images URL of the Post Gallery

Your code is going to cause a fatal error as you have defined a function inside a loop, and therefore are redefining the function at every iteration thereafter. That is part of the problem. Take a look at cleaned up code:

function pw_show_gallery_image_urls( $content ) {

  global $post;

  // Only do this on singular items
  if( ! is_singular() )
      return $content;

  // Make sure the post has a gallery in it
  if( ! has_shortcode( $post->post_content, 'gallery' ) )
      return $content;

  // Retrieve the first gallery in the post
  $gallery = get_post_gallery_images( $post );

  $image_list="<ul>";

  // Loop through each image in each gallery
  foreach( $gallery as $image_url ) {
      $image_list .= '<li>' . '<img src="' . $image_url . '">' . '</li>';
  }

  $image_list .= '</ul>';

  // Append our image list to the content of our post
  $content .= $image_list;

  return $content;
}
add_filter( 'the_content', 'pw_show_gallery_image_urls' );

if ( have_posts() ) {
  while ( have_posts() ) {
    the_post(); 

  } // end while
} // end if

Note the Loop at the bottom. You have added a filter to the_content but there is nothing in the Loop that would apply that filter, or even print any content at all. You need:

if ( have_posts() ) {
  while ( have_posts() ) {
    the_post(); 
    the_content(); // minimum
  } // end while
} // end if