Following V4l3ri4's link and the links that spawned from there, I came up with the following bare bones solution which shrinks the text within the TextView continually until it fits width-wise in the TextView:
public class FontFitTextView extends TextView
{
  private float maxTextSizePx;
  public FontFitTextView(Context context)
  {
    super(context);
    initialise();
  }
  public FontFitTextView(Context context, AttributeSet attrs)
  {
    super(context, attrs);
    initialise();
  }
  public FontFitTextView(Context context, AttributeSet attrs, int defStyle)
  {
    super(context, attrs, defStyle);
    initialise();
  }
  /** Sets the maximum text size as the text size specified to use for this View.*/
  private void initialise()
  {
    maxTextSizePx = getTextSize(); 
  }
  /** Reduces the font size continually until the specified 'text' fits within the View (i.e. the specified 'viewWidth').*/
  private void refitText(String text, int viewWidth)
  { 
    if (viewWidth > 0)
    {
      TextPaint textPaintClone = new TextPaint();
      textPaintClone.set(getPaint());
      int availableWidth = viewWidth - getPaddingLeft() - getPaddingRight();
      float trySize = maxTextSizePx;
      // note that Paint text size works in px not sp
      textPaintClone.setTextSize(trySize); 
      while (textPaintClone.measureText(text) > availableWidth)
      {
        trySize--;
        textPaintClone.setTextSize(trySize);
      }
      setTextSize(TypedValue.COMPLEX_UNIT_PX, trySize);
    }
  }
  @Override
  protected void onTextChanged(final CharSequence text, final int start, final int lengthBefore, final int lengthAfter)
  {
    super.onTextChanged(text, start, lengthBefore, lengthAfter);
    refitText(text.toString(), getWidth());
  }
  @Override
  protected void onSizeChanged(int w, int h, int oldw, int oldh)
  {
    super.onSizeChanged(w, h, oldw, oldh);
    if (w != oldw)
      refitText(getText().toString(), w);
  }
}
An example usage is as follows:
<view
  class="com.mycompany.myapp.views.FontFitTextView"
  android:layout_width="0dp"
  android:layout_height="wrap_content"
  android:layout_weight="1"
  android:singleLine="true" />
I realise this implementation can be optimised and extended but just trying to show a bare bones solution for you to extend or modify as required.
Oh and if you need a Button which shrinks text to fit instead of a TextView, simply use the code exact as above but extend Button instead of TextView.