android:gravity="right|center_vertical"

aligns your TextView to the right. How about center_horizontal?

And your LinearLayouts have the same ID, if there is not more code, then I would delete one of them.

I found none of the answers work exactly as the OP intended, but the RelativeLayout solution was the closest one. Except that the TextView wasn't centered. But, with layout_centerInParent=true, works like a charm:

<RelativeLayout
    android:layout_width="match_parent"
    android:layout_height="wrap_content">
    <TextView 
         android:id="@+id/TextView00"
         android:layout_width="wrap_content"
         android:layout_height="wrap_content"
         android:textColor="#FFFFFF"
         android:textSize="20px"
         android:textStyle="bold"
         android:text="Something"
         android:layout_centerInParent="true" />

    <Button 
         android:id="@+id/btnCalls"
         android:drawableLeft="@drawable/icon"
         android:layout_width="wrap_content"
         android:layout_height="wrap_content"
         android:layout_alignParentRight="true"/>
</RelativeLayout>

you have used android:gravity="right|center_vertical" in your textview. Use android:gravity="center".

Try to add empty View inside horizontal LinearLayout between element that you want see left and element that you want to see right, e.g.:

    <LinearLayout
        android:layout_width="fill_parent"
        android:layout_height="wrap_content" >

        <Button
            android:layout_width="wrap_content"
            android:layout_height="wrap_content" />

        <View
            android:layout_width="0dp"
            android:layout_height="0dp"
            android:layout_weight="1"
            />                

        <Button
            android:layout_width="wrap_content"
            android:layout_height="wrap_content" />

    </LinearLayout>

define layout weight for TextView and LinearLayout

set android:layout_weight="1" to TextView

set android:layout_weight="0" to LinearLayout

and also change the layout_width="wrap_content" to LinearLayout and set layout_gravity="center" to TextView

I think,gravity means the position of things inside the component(for textview,it means the text you have given)in component where layout_gravity means position of component itself in the whole layout.So...

Try Using:

<?xml version="1.0" encoding="utf-8"?>
    <LinearLayout android:id="@+id/LinearLayout01"
        android:layout_width="fill_parent"
        android:layout_height="fill_parent"
        xmlns:android=""
        android:orientation="vertical"
        android:background="#FFFFFF"
        >
        <LinearLayout android:id="@+id/LinearLayout01"
            android:layout_width="fill_parent"
            android:layout_height="wrap_content"
            xmlns:android=""
            android:orientation="horizontal"
            android:background="#E30000"
            android:layout_gravity="right"
            android:gravity="center_vertical|right"                    
            >
                <LinearLayout
                    android:layout_width="wrap_content"
                    android:layout_height="wrap_content"
                    android:layout_gravity="right|center_vertical"
                    android:gravity="center_vertical|center" 
                    android:layout_weight="0.9"
                >
                    <TextView 
                        android:id="@+id/TextView00"
                        android:layout_width="wrap_content"
                        android:layout_height="wrap_content"
                        android:textColor="#FFFFFF"
                        android:textSize="20px"
                        android:textStyle="bold"
                        android:text="Something"
                        android:gravity="center_horizontal"
                        android:layout_gravity="center_horizontal|center_vertical" 

                    />
                </LinearLayout>
                <LinearLayout
                    android:layout_width="wrap_content"
                    android:layout_height="wrap_content"
                    android:layout_gravity="right|center_vertical"
                    android:gravity="center_vertical|right"                    
                    android:paddingTop="5dip"
                    android:layout_weight="0.1"
                >
                    <Button 
                        android:id="@+id/btnCalls"
                        android:layout_width="wrap_content"
                        android:layout_height="wrap_content"
                        android:drawableLeft="@drawable/icon"
                        android:layout_gravity="center_vertical"
                        android:gravity="center"                                    
                    />
                </LinearLayout>
        </LinearLayout>
    </LinearLayout>

Just use RelativeLayout, and wallaaa!!!

<RelativeLayout
    android:layout_width="match_parent"
    android:layout_height="wrap_content">
    <TextView 
         android:id="@+id/TextView00"
         android:layout_width="wrap_content"
         android:layout_height="wrap_content"
         android:textColor="#FFFFFF"
         android:textSize="20px"
         android:textStyle="bold"
         android:text="Something"
         android:gravity="center" />

    <Button 
         android:id="@+id/btnCalls"
         android:drawableLeft="@drawable/icon"
         android:layout_width="wrap_content"
         android:layout_height="wrap_content"
         android:layout_alignParentRight="true"/>
</RelativeLayout>

The solution for this is- android:gravity="center_vertical"

Normally you'll also have to take care of its parent's position too to make sure it works.