Don't believe you can, users can still remove their image from profile edit for instance or from image upload page. You can try hiding with CSS, but you're likely going to need to make core edits to cb.menu.php (which I can not suggest, sorry).
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Thanks for taking the time to respond. You have helped me with some of your past posts. I managed to work this one out. I wanted to do this because I use CB on an internal intranet and I have made avatars a required field where employees either need to upload their own avatar or choose from an employee gallery. Once they have an avatar I do want them to be able to remove it, only update it if necessary. To do this I commented out some lines of code in the cb.menu.php file located at components\com_comprofiler\plugin\user\plug_cbmenu
Below is a list of the lines I commented out.
LINE 1029 //'_UE_DELETE_AVATAR' => _UE_DELETE_AVATAR,
LINE 1030 //'_UE_MENU_DELETE_AVATAR_DESC' => _UE_MENU_DELETE_AVATAR_DESC
LINE 1037 //'_UE_DELETE_AVATAR' =>_UE_MOD_MENU_DELETE_AVATAR,
LINE 1038 //'_UE_MENU_DELETE_AVATAR_DESC' => _UE_MOD_MENU_DELETE_AVATAR_DESC
LINE 1051 //if($user->avatar!='' && $user->avatar!=null) {
LINE 1052 //$mi = array(); $mi["_UE_MENU_EDIT"]["_UE_DELETE_AVATAR"]=null;
LINE 1053 //$this->_addMenuItem( $mi, $menuTexts,cbSef($ue_deleteavatar_url), "",
LINE 1054 //"<img src=\"".$adminimagesdir."delavatar.gif\" alt='' />","", $menuTexts,"" );
LINE 1055 //}
Lastly comment out the following lines of components\com_comprofiler\plugin\user\plug_cbcore\cb.core.php
This will remove the delete image option from the dropdown list of the Edit Your Details/Portrait Tab.