Define a new policy to disable clearing the browsing history from the UI.

chrome/browser/resources/options/clear_browser_data_overlay.html

 <div id="clear-browser-data-overlay" class="page not-resizable" hidden>
   <div class="close-button"></div>
   <h1 i18n-content="clearBrowserDataOverlay"></h1>
   <!-- NOTE: Make sure there's not whitespace between <div></div> for
        #clear-browser-data-info-banner as it's styled with :empty. -->
   <div id="clear-browser-data-info-banner"></div>
   <div id="cbd-content-area" class="content-area">
     <span i18n-content="clearBrowserDataLabel"></span>
     <select id="clear-browser-data-time-period"
         i18n-options="clearBrowserDataTimeList"
         pref="browser.clear_data.time_period"
         data-type="number">
     </select>
     <div id="clear-data-checkboxes">
       <div id="delete-browsing-history-container" class="checkbox">
         <label>
           <input id="delete-browsing-history-checkbox"
               pref="browser.clear_data.browsing_history" type="checkbox"
               aria-controls="delete-browsing-history-counter">
           <span>
             <span i18n-content="deleteBrowsingHistoryCheckbox"></span>
+            <span class="controlled-setting-indicator"
+                pref="history.deleting_enabled">
+            </span>
             <span class="clear-browser-data-counter" role="note"
                 aria-live="polite" id="delete-browsing-history-counter"></span>
           </span>
         </label>
       </div>
       <div id="delete-download-history-container" class="checkbox">
         <label>
           <input id="delete-download-history-checkbox"
               pref="browser.clear_data.download_history" type="checkbox">
           <span i18n-content="deleteDownloadHistoryCheckbox"></span>
+          <span class="controlled-setting-indicator"
+              pref="history.deleting_enabled">
+          </span>
         </label>
       </div>
       <div id="delete-cookies-container" class="checkbox">
         <label>
           <input id="delete-cookies-checkbox"
               pref="browser.clear_data.cookies" type="checkbox">
           <span i18n-content="deleteCookiesFlashCheckbox"
               class="clear-plugin-lso-data-enabled"></span>
           <span i18n-content="deleteCookiesCheckbox"
               class="clear-plugin-lso-data-disabled"></span>
@@ skipping 80 matching lines @@
     <p id="clear-browser-data-history-footer"
         i18n-values=".innerHTML:clearBrowserDataHistoryFooter" hidden></p>
     <p id="clear-browser-data-general-footer">
       <span><!--This is filled by JavaScript--></span>
       <a id="clear-browser-data-footer-learn-more-link" hidden
           target="_blank" i18n-content="learnMore"
           i18n-values="href:clearBrowsingDataLearnMoreUrl"></a>
     </p>
   </div>
 </div>