I finally solved this myself. If anyone else is having this problem, here is my solution:

I created a new method:

public function curl_del($path)
{
    $url = $this->__url.$path;
    $ch = curl_init();
    curl_setopt($ch, CURLOPT_URL, $url);
    curl_setopt($ch, CURLOPT_CUSTOMREQUEST, "DELETE");
    $result = curl_exec($ch);
    $httpCode = curl_getinfo($ch, CURLINFO_HTTP_CODE);
    curl_close($ch);

    return $result;
}

Update 2

Since this seems to help some people, here is my final curl DELETE method, which returns the HTTP response in JSON decoded object:

/**
 * @desc    Do a DELETE request with cURL
 *
 * @param   string $path   path that goes after the URL fx. "/user/login"
 * @param   array  $json   If you need to send some json with your request.
 *                         For me delete requests are always blank
 * @return  Obj    $result HTTP response from REST interface in JSON decoded.
 */
public function curl_del($path, $json = '')
{
    $url = $this->__url.$path;
    $ch = curl_init();
    curl_setopt($ch, CURLOPT_URL, $url);
    curl_setopt($ch, CURLOPT_CUSTOMREQUEST, "DELETE");
    curl_setopt($ch, CURLOPT_POSTFIELDS, $json);
    curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
    $result = curl_exec($ch);
    $result = json_decode($result);
    curl_close($ch);

    return $result;
}