stringbuilder data not showing properly

This is due to operator precedence. Your code is equivalent to:

if (("<tr><td>Call Time: " + searchResult[i].CreatedDate) != null)
{
    sb.Append(Convert.ToDateTime(searchResult[i].CreatedDate).ToString("hh:mm tt"));
}
else
{
    sb.Append("" + "</td></tr>");
}

Since anything appended to a non-null string will always produce a non-null value, you're always going to end up in the first branch.
Ignore that line - it's totally wrong.

Either split the code in to multiple lines:

sb.Append("<tr><td>Call Time: ");
if (searchResult[i].CreatedDate != null)
{
    sb.Append(Convert.ToDateTime(searchResult[i].CreatedDate).ToString("hh:mm tt"));
}
sb.Append("</td></tr>");

or add parentheses to force the precedence:

sb.Append("<tr><td>Call Time: " + (searchResult[i].CreatedDate != null ? Convert.ToDateTime(searchResult[i].CreatedDate).ToString("hh:mm tt") : "") + "</td></t>");

or use AppendFormat:

sb.AppendFormat("<tr><td>Call Time: {0:hh:mm tt}</td></t>", searchResult[i].CreatedDate);

Brackets!

//this
sb.Append("<table><tbody><tr><td>Call Time: " + searchResult[i].CreatedDate != null ? Convert.ToDateTime(searchResult[i].CreatedDate).ToString("hh:mm tt") : "" + "</td></tr></tbody></table>");
//is like this
sb.Append(("<table><tbody><tr><td>Call Time: " + searchResult[i].CreatedDate != null ? Convert.ToDateTime(searchResult[i].CreatedDate).ToString("hh:mm tt") : "") + "</td></tr></tbody></table>");

//you need this
sb.Append("<table><tbody><tr><td>Call Time: " + (searchResult[i].CreatedDate != null ? Convert.ToDateTime(searchResult[i].CreatedDate).ToString("hh:mm tt") : "") + "</td></tr></tbody></table>");

Operator precedence is the key here...
Try this:

object o = null;
string s = "1" + o == null ? "2" : "3" + "4";
string t = "1" + o != null ? "2" : "3" + "4";

And compare the values of s and t

s: "34"
t: "2"

The reason for this is that the "+" has a higher precedence than "?" - so the beginning bit is swallowed.
Adding brackets to override the precedence:

object o = null;
string s = "1" + (o == null ? "2" : "3") + "4";
string t = "1" + (o != null ? "2" : "3") + "4";

Gives the values you expect:

s: "124"
t: "134"

So add the brackets to your code, and it should work:

sb.Append("<tr><td>Call Time: " + (searchResult[i].CreatedDate != null ? Convert.ToDateTime(searchResult[i].CreatedDate).ToString("hh:mm tt") : "") + "</td></tr>");

The problem looks simple like hell. First of all, it makes no sense to ask questions about StringBuilder behavior if you are showing only one string in your code sample. Believe me, StringBuilder does not do anything wrong; it simply adds the string; there is no point in seeing what's happening in its instance. Instead, look at your string.

How can you try to make your code in one like and complain about difficulties in debugging at the same time. If you really want to debug, write your string step by step and have a separate variable for the resulting string.

Never repeat any code. Therefore, calculate searchResult[i].CreatedDate separately, assign the result to a different variable. Always, even in your working code, not just for debugging. And then assign the string in the middle, the one you obtain from ToString and the conditional operator, to a separate string variable. Inspect it under the debugger. Let's say, this is the variable myMiddleString (don't name it like that in real code, of course; always use semantic names).

And finally, what's the point of using so performance-improving tool as StringBuilder, if you are using such a performance-killing thing as multiple string concatenation? Of course, this performance leak is minor compared to the StringBuilder gain, but nevertheless; you can also improve readability. Don't do it! Use

string myStringToAdd = string.Format("<tr><td>Call Time: {0}</td></tr>", myMiddleString);

And then you can see what's going on in the debugger. StringBuilder is irrelevant here.
Now, about HTML: from your sample, it's apparent that you are creating table rows with one cell each; it does not seem to make much sense.

Otherwise, the idea of your code looks correct. Your particular problem is explained in other answers, please see them.

—SA