Ping Url or DNS Name From Lambda Function

I have a Lambda function, not running in a VPC, that does some simple HTTP, TCP, and ICMP health checks. The HTTP/S and TCP checks work just fine, however, when I try to ping a site or host name (like www.amazon.com or www.google.com), I get a Timed Out response. Is this because of the "statelessness" of ICMP and the reply ECHO responses simply aren't being allowed back through to the Lambda function? I don't want to run this in a VPC since the service is independent of other resources in the AWS account. Any ideas?

My code is as follows, and works running locally:

    using (Ping Pinger = new Ping())
    {
      try
        {
          PingReply Reply = await Pinger.SendPingAsync(Path, 500);

          if (Reply.Status == IPStatus.Success)
          {
            Success = true;
            Message = $"[INFO] : {Path} via PING is up!";
          }
          else
          {
            Message = $"[ERROR] : {Path} via PING is down with status: {Reply.Status.ToString()}.";
           }
         }
         catch (Exception e)
         {
           Message = $"[ERROR] : {Path} via PING failed with an unexpected error:\r\n{JsonConvert.SerializeObject(e)}.";
         }
       }