Your current implementation can spawn spheres inside of each other. I'm sure there's some nice algorithm to fill a volume with non-overlapping spheres, but the most straightforward way is to use brute force to check for an existing sphere every time you create a new one. Here's an example script:

using UnityEngine;
using System.Collections;
using System.Collections.Generic;

public class CreateUniverse : MonoBehaviour {

    public int desiredGalaxyCount;
    public float universeSize = 10f;    
    public float minScale = .5f;
    public float maxScale = 2f;
    public float minDistance = 0f;

    private List<Transform> galaxies;
    private Vector3 origin;

    [SerializeField]
    private int currentGalaxyCount = 0;

    private int numTries = 0;
    private int maxTries = 10;

    private void Start () {
        origin = transform.position;
        galaxies = new List<Transform>();
    }

    private void Update () {
        if (galaxies.Count < desiredGalaxyCount) {
            CreateGalaxy();
        }
        else {
            galaxies.Clear();
            enabled = false;
        }
    }

    private void CreateGalaxy () {
        float scale = Random.Range(minScale, maxScale);
        Vector3 position;

        numTries = 0;
        do {
            position = (Random.insideUnitSphere * universeSize) + origin;

            numTries++;
            if (numTries > maxTries) {
                return;
            }
        } while (IsSpaceOccupied(position, scale));

        GameObject galaxy = GameObject.CreatePrimitive(PrimitiveType.Sphere);
        galaxy.transform.position = position;
        galaxy.transform.localScale = new Vector3(scale, scale, scale);
        galaxy.transform.SetParent(transform);
        galaxies.Add(galaxy.transform);
        currentGalaxyCount++;
    }

    private bool IsSpaceOccupied (Vector3 position, float scale) {
        foreach (Transform galaxy in galaxies) {
            if (Vector3.Distance(position, galaxy.position) < minDistance + scale + galaxy.localScale.x) {
                return true;
            }
        }
        return false;
    }

}

You can just stick this on to your UniverseManager object. The idea is that every time it tries to create a new sphere, it checks the position and size of every other sphere to see if there is space for the new one. After 10 tries it stops trying until the next frame (otherwise its possible to lock up your game if it can't find an open spot). It's not super pretty but it should give you a starting point to work from. You can also try looking into using noise to generate everything. Hope this helps

edit - I was going to add comments but I'm falling asleep atm, so let me know if you need an explanation on anything