You could try making your div float:
.thatSetsABackgroundWithAnIcon{
float:left;
}
Alternativelly, use inline-block:
.thatSetsABackgroundWithAnIcon{
display:inline-block;
}
Working example of the inline-block method:
table,_x000D_
th,_x000D_
td {_x000D_
border: 1px solid black;_x000D_
}
_x000D_
<table>_x000D_
<tr>_x000D_
<td>_x000D_
<div style="border:1px solid red; height:100%; display:inline-block;">_x000D_
I want cell to be the full height_x000D_
</div>_x000D_
</td>_x000D_
<td>_x000D_
This cell_x000D_
<br/>is higher_x000D_
<br/>than the_x000D_
<br/>first one_x000D_
</td>_x000D_
</tr>_x000D_
</table>
_x000D_