Just set the line-height for both. No need to work on inline-block display, they're both inline elements anyways. The problem with your fiddle is that you have forgot to insert the [dot] from the class form-group and if you're using Boostrap, you did not include the external CSS library.
Here is a working fiddle - http://jsfiddle.net/xeqLsdo6/2/
And on this page is a working scenario based on BS and your desired code. From here, you could work on line-heights and other stuff.
I don't really know what you mean by horizontal alignment. Is it about both label and select should stay in the middle of the screen or that the option element should be centered inside the select element?
Other than that, more explanations would not hurt anyone. Just firing up a code somewhere won't help you much, because you could forget things and a solution becomes less probable.
#state {
display: inline;
width: auto;
}
<link href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/css/bootstrap.min.css" rel="stylesheet"/>
<div class="form-group col-lg-6 col-sm-6">
<label for="state">State:<span class="star">*</span></label>
<select required id="state" name="state" title="State" class="form-control">
<option value="">--None--</option>
<option value="NSW">NSW</option>
<option value="VIC">VIC</option>
<option value="QLD">QLD</option>
<option value="WA">WA</option>
<option value="SA">SA</option>
<option value="TAS">TAS</option>
<option value="NT">NT</option>
<option value="ACT">ACT</option>
<option value="Other">Other</option>
</select>
</div>