How to design 4-bit comparator using the below described logic?
Yes, this is certainly possible. 

From your question it is not clear whether you want a 4-bit comparator that outputs only equality, or the three conclusions as shown for your 1-bit comparator.

This smells of homework, and as a rule we don't give full answers to such questions, but we can gently steer you in the right direction.

Consider the equality. Your 1) text indicates that for a 0 output the conclusion must be 0, for a 1 output the conclusion must be the 'other' input (from the lower bis).


which logic gate does this describe?
how would you connect this gate to the single bit comparator (you will need one extra input, the 'lower bits are equal' input (often called 'cascade')) Draw the diagram of this one-bit-comparator-with-cascade-input.
Now draw the final 4-bit comparator.