New 4.6 UI question: How do you resize a panel relative to the screen but remain height/width constrained?

Another method is a custom script that I made, just keep in mind that the object should always be set to the highest screen resolution so the UI does not blur.

 using UnityEngine;
 using System.Collections;
 
 [AddComponentMenu("UI/UI Resizer")]
 public class UIResizer : MonoBehaviour
 {
 
     // The resolution that the UI was initially designed for.
     [SerializeField]
     Vector2 targetResolution = new Vector2(1920, 1080);
 
     // The Transform that should be resized, if not given then itself.
     [SerializeField]
     Transform UIObject;
 
     void Start()
     {
         if(!UIObject)
             UIObject = transform;
         SizeResolution (new Vector2 (Screen.width, Screen.height));
     }
 
     public void SizeResolution(Vector2 newResolution)
     {
         Vector3 scale = UIObject.localScale;
         scale.x = newResolution.x / targetResolution.x;
         scale.y = newResolution.y / targetResolution.y;
         UIObject.localScale = scale;
     }
 
 }

Actually isn't necessary anymore do extra-code to make the UI fit in screen. Usign "Canvas Scaler" component, you can select the "Scale With Screen Size" and define the best screen size for Height and move the slider entirely to height. you will have you canvas scaling and not deforming the elements in screen.

Of course you still needing have some work to anchor the UI on left/right to perform correct Aspect Ratio resize on multiple screens.

The best solution for this is correct configuration of the Canvas:

As I see the best practice is:

1) to set render mode to "Screen Space - Camera"

2) Plane distance to 2-10

3. Assign one additional Canvas Scaler to your Canvas and set the following settings:

3.a: Ui ScaleMode:Scale with screen Size

3.b: Reference Resolution 2500x1440 (or your max srceen size that you planning to develop for)

3.c: Match mode: Height (1)

4) Set width and height of children with some stock

5) In all texts change Vertical Overflow to "Overflow"

this will solve lot of problems like:

1) large objects on scene

2) pixelization of objects edges

3) Absent ability to add ParticleSystem to your UI screen (sometimes needed in mobile menus games)

4) Absent ability to resize a panel relative to the screen but remain height/width constrained

5) Text hide on font resize

But in your case you can use only 3rd (highlited)

Just go to the Canvas Scaler component that belongs to your canvas and set UI Scale Mode to Scale With Screen Size then set the Reference Resolution!

That is all!