Modifying memory of an integrated circuit
Is it possible to make an integrated circuit in which the 
  memory can only be modified and read through the microprocessor and
  not directly?


That depends on your definition of 'only'. 

Smart-cards have several layers of protection which makes it very hard to open the chip, bond some wires on the transistors and then read/write them using external equipment.

However, given enough time and money it is still possible to do this. The trick is to make such attempts so expensive, that it is not economical feasible to attack the memory that way.

Of cause technology advances, and with new attacks come new countermeasures. That's why smart-cards manufacturers are constantly improving their designs based on the latest research.

--- Edit ---

This talk from the 30c3 hacking conference talks about the protection you'll find in smart-cards and what is involved in reading out the memory. (includes cool stuff like lasers!)

https://media.ccc.de/browse/congress/2013/30C3_-5459-__-_saal_1_-201312282145-_security_of_the_ic_backside_-_nedos.html#video