I have edited the answer given by @Jaswanth Manigundan to make it extend from android.support.v7.widget.AppCompatButton. Have a look at the below scripts:
FontFitTextView.java
import android.content.Context;
import android.content.res.TypedArray;
import android.util.AttributeSet;
import android.util.TypedValue;
public class FontFitTextView extends android.support.v7.widget.AppCompatButton
{
private static final float THRESHOLD = 0.5f;
private enum Mode { Width, Height, Both, None }
private int minTextSize = 1;
private int maxTextSize = 50;
private Mode mode = Mode.None;
private boolean inComputation;
private int widthMeasureSpec;
private int heightMeasureSpec;
public FontFitTextView(Context context) {
    super(context);
}
public FontFitTextView(Context context, AttributeSet attrs) {
    this(context, attrs, 0);
}
public FontFitTextView(Context context, AttributeSet attrs, int defStyle) {
    super(context, attrs, defStyle);
    TypedArray tAttrs = context.obtainStyledAttributes(attrs, R.styleable.FontFitTextView, defStyle, 0);
    maxTextSize = tAttrs.getDimensionPixelSize(R.styleable.FontFitTextView_maxTextSize, maxTextSize);
    minTextSize = tAttrs.getDimensionPixelSize(R.styleable.FontFitTextView_minTextSize, minTextSize);
    tAttrs.recycle();
}
private void resizeText() {
    if (getWidth() <= 0 || getHeight() <= 0)
        return;
    if(mode == Mode.None)
        return;
    final int targetWidth = getWidth();
    final int targetHeight = getHeight();
    inComputation = true;
    float higherSize = maxTextSize;
    float lowerSize = minTextSize;
    float textSize = getTextSize();
    while(higherSize - lowerSize > THRESHOLD) {
        textSize = (higherSize + lowerSize) / 2;
        if (isTooBig(textSize, targetWidth, targetHeight)) {
            higherSize = textSize;
        } else {
            lowerSize = textSize;
        }
    }
    setTextSize(TypedValue.COMPLEX_UNIT_PX, lowerSize);
    measure(widthMeasureSpec, heightMeasureSpec);
    inComputation = false;
}
private boolean isTooBig(float textSize, int targetWidth, int targetHeight) {
    setTextSize(TypedValue.COMPLEX_UNIT_PX, textSize);
    measure(0, 0);
    if(mode == Mode.Both)
        return getMeasuredWidth() >= targetWidth || getMeasuredHeight() >= targetHeight;
    if(mode == Mode.Width)
        return getMeasuredWidth() >= targetWidth;
    else
        return getMeasuredHeight() >= targetHeight;
}
private Mode getMode(int widthMeasureSpec, int heightMeasureSpec) {
    int widthMode = MeasureSpec.getMode(widthMeasureSpec);
    int heightMode = MeasureSpec.getMode(heightMeasureSpec);
    if(widthMode == MeasureSpec.EXACTLY && heightMode == MeasureSpec.EXACTLY)
        return Mode.Both;
    if(widthMode == MeasureSpec.EXACTLY)
        return Mode.Width;
    if(heightMode == MeasureSpec.EXACTLY)
        return Mode.Height;
    return Mode.None;
}
@Override
protected void onMeasure(int widthMeasureSpec, int heightMeasureSpec) {
    super.onMeasure(widthMeasureSpec, heightMeasureSpec);
    if(!inComputation) {
        this.widthMeasureSpec = widthMeasureSpec;
        this.heightMeasureSpec = heightMeasureSpec;
        mode = getMode(widthMeasureSpec, heightMeasureSpec);
        resizeText();
    }
}
protected void onTextChanged(final CharSequence text, final int start, final int before, final int after) {
    resizeText();
}
@Override
protected void onSizeChanged(int w, int h, int oldw, int oldh) {
    if (w != oldw || h != oldh)
        resizeText();
}
public int getMinTextSize() {
    return minTextSize;
}
public void setMinTextSize(int minTextSize) {
    this.minTextSize = minTextSize;
    resizeText();
}
public int getMaxTextSize() {
    return maxTextSize;
}
public void setMaxTextSize(int maxTextSize) {
    this.maxTextSize = maxTextSize;
    resizeText();
}
}
style.xml:
<resources>
<declare-styleable name="FontFitTextView">
    <attr name="minTextSize" format="dimension" />
    <attr name="maxTextSize" format="dimension" />
</declare-styleable>
</resources>
And in my activity_main.xml, I used the FontFitTextView tag instead of Button:
<com.example.myname.testsidebar.FontFitTextView
            android:id="@+id/userDetailsBtn"
            android:layout_width="match_parent"
            android:layout_height="0dp"
            android:drawableTop="@drawable/user_details"
            android:gravity="center"
            android:background="#00FFFFFF"
            android:paddingTop="10sp"
            android:textSize="15sp"
            android:text="Your Details"
            android:drawableTint="#ED1B2F"
            android:textColor="#797369"
            android:layout_weight="1"
            android:layout_gravity="center"/>
Hope it helps.