We show that a blank-tape-halt-checker can not exist by reduction from the Halting Problem.

Assume that a blank-tape-halt-checker BT(P) does exist.
We build a program named HC, which takes a program P and a string w as input, as follows:

  HC(P,w):
     (1) Build a Turing Machine that takes no inputs. It first writes the string w
         to its blank tape, and then runs P on that tape. Call this no-input
         turing machine TM. Note that TM effectively runs P on w.

     (2) Call our blank-tape-halt-checker on this TM:  BT( TM ) 

     (3) If BT(TM) reports that TM halts, halt and output "Yes"

     (4) If BT(TM) reports that TM does not halt, halt and output "No"

As long as BT exists, this constructed HC is a legitimate program. 
All of the steps are computable -- writing a single, known string to a 
blank tape, running a turing machine's program, and conditional-checking.

However, note that HC(P,w) is a halt checker!

    (A) If P halts on w, HC(P,w) returns "Yes" 
           (because TM will halt on no input, so BT(TM) returns "Yes")

    (B) If P does not halt on w, HC(P,w) returns "No" 
           (because TM will not halt on no input, so BT(TM) returns "No")

Since a halt-checker can not exist, we have reached a contradiction. 
Thus, our original assumption that the blank-tape-halt-checker exists was false. 
A blank-tape-halt-checker also can not exist.