Is it possible to stop a theme activation when a certain plugin is not activated

Using after_switch_theme will activate the theme (which is fine as we want to run the check within the context of the new theme).

If the dependencies are not fulfilled ($missing_dependencies = true;) we're instantly switching back to the theme previously used (passed via after_switch_theme and $oldtheme).

add_action( 'after_switch_theme', 'check_theme_dependencies', 10, 2 );
function check_theme_dependencies( $oldtheme_name, $oldtheme ) {
  if ( $missing_dependencies ) :

    // Update default admin notice: Theme not activated.
    add_filter( 'gettext', 'update_activation_admin_notice', 10, 3 );

    // Custom styling for default admin notice.
    add_action( 'admin_head', 'error_activation_admin_notice' );

    // Switch back to previous theme.
    switch_theme( $oldtheme->stylesheet );
      return false;

  endif;
}

function update_activation_admin_notice( $translated, $original, $domain ) {
    // Strings to translate.
    $strings = array(
        'New theme activated.' => 'Theme not activated.'
    );

    if ( isset( $strings[$original] ) ) {
        // Translate but without running all the filters again.
        $translations = get_translations_for_domain( $domain );
        $translated = $translations->translate( $strings[$original] );
    }

    return $translated;
}

function error_activation_admin_notice() {
  echo '<style>#message2{border-left-color:#dc3232;}</style>';
}

You can use the snippet above within your functions.php, but make sure that after_switch_theme is called before the required dependencies.

Update: There does not seem to be an easy way to prevent the admin notice triggered via the switch_theme function. Overwriting the message via the gettext filter seems to be a good workaround.

Thanks to @alpipego for telling me how to overwrite strings in WordPress Core :)

Further reading: Changing Core WordPress Strings

I don't know how to prevent the theme switch (though there may well be a way), however I can think of a workaround.

after_switch_theme passes the old theme name through...

do_action( 'after_switch_theme', $old_theme->get( 'Name' ), $old_theme );

... so you should be able to check for the plugin on that action and then reset to the old theme if the plugin is not active/present. The switch_theme() function will allow you to programatically switch themes. Just pass it a stylesheet slug. I just answered a related question that will problably help with this problem too.

I don't think it'll prevent anyone from activating a theme, but here's a class for attaching plugin dependencies to themes (or to other plugins): https://github.com/thomasgriffin/TGM-Plugin-Activation

At the very least, it adds all the UI notices and nags so that the administrator knows they're missing something. It even adds links to install the missing dependencies. Pretty cool.

Couldn't you just include the plugin with the theme, if it's one you're developing? The disadvantages of this approach include not updating the plugin separately, but if your theme is dependent on a particular version of a particular plugin, it may work just fine, as long as you patch the plugin you require.