Outline on Texture2D

I'm creating a minimap, but I want to add a little outline effect, just like in this answer: http://answers.unity3d.com/questions/1036262/shaders-offset-texture-coordinates-by-a-single-pix.html

But without shaders, because this texture will be saved as a png file, then use it as a sprite inside a UI Image.

This is what I have done: Looks good, but how can I add a simple 1 pixel outline inside the texture?

For extra details, here is my script:

 using UnityEngine;
 using System.Collections;
 using System.IO;
 
 [RequireComponent(typeof(Camera))]
 public class MiniMap : MonoBehaviour {
 
     private Camera cam;
 
     void Awake () {
         cam = GetComponent<Camera> ();
 
         CreateMiniMap ();
 
         Destroy (gameObject);
     }
 
     void CreateMiniMap () {
         RenderTexture rt = new RenderTexture (512, 512, 16, RenderTextureFormat.ARGB32); //Create a render texture
         Texture2D minimap = new Texture2D (rt.width, rt.height, TextureFormat.ARGB32, false);
 
         cam.targetTexture = rt; 
         cam.Render ();
         RenderTexture.active = rt;
 
         minimap.ReadPixels (new Rect (0, 0, rt.width, rt.height), 0, 0); //Render texture to Texture2D
         minimap.Apply ();
         RemoveColor (cam.backgroundColor, Color.clear, minimap); //Removes the background of the Texture2D
         FillImage (Color.black, minimap);
 
         cam.targetTexture = null;
         RenderTexture.active = null;
         DestroyImmediate (rt);
 
         if (!Directory.Exists (Application.dataPath + "/MiniMaps")) {
             Directory.CreateDirectory(Application.dataPath + "/MiniMaps");
         } //Creates a directory to save the minimap
 
         File.WriteAllBytes (Application.dataPath + "/MiniMaps/" + GameManager.map.name + "_Minimap.png", minimap.EncodeToPNG ());
     } //Save as a PNG file
 
     void RemoveColor (Color color, Color newColor, Texture2D tex) {
         Color[] pixels = tex.GetPixels (0, 0, tex.width, tex.height, 0);
 
         for (int i = 0; i < pixels.Length; i++) {
             if (pixels[i] == color)
                 pixels[i] = newColor;
         }
 
         tex.SetPixels (0, 0, tex.width, tex.height, pixels, 0);
         tex.Apply ();
     }
 
     void FillImage (Color fillColor, Texture2D tex) {
         Color[] pixels = tex.GetPixels (0, 0, tex.width, tex.height, 0);
         
         for (int i = 0; i < pixels.Length; i++) {
             if (pixels[i] != Color.clear)
                 pixels[i] = fillColor;
         }
         
         tex.SetPixels (0, 0, tex.width, tex.height, pixels, 0);
         tex.Apply ();
     }
 }