You're passing fundamental types by const reference. These are better off just being passed by value. So you'd get things like
explicit address(uint32_t value);
reference operator[](int index) noexcept(false);
Your prefix increment and decrement operators should return a reference to the incremented value.
address operator++();
address operator--();
This will allow expressions like addr = ++other_addr;. (Note that, since you're in the address class, you can just name the class, you don't need to specify scope with ::ip::address).
Your postfix increment and decrement operators have a bug, because they return a reference to a local variable. The return types should be a value.
address operator++(int);
address operator--(int);
For readability and clarity, expressions mixing shifts and bit masking should use parentheses:
data_[0] = (value >> 24) & 0xFF;