Can I use MCP23008 with G6K-2F-Y relay?
From Microchip's data sheet:

The I/O pin is not spec'd for more than 3mA as a current source. It is anyone's guess how much voltage is dropped below the supply for larger currents, like 20mA.... At 3mA, you're guaranteed 4.3V. It is a very weak switch for coil current.
More promising is sink current - it is a better switch pulling down compared to pulling up. Again, it is anyone's guess how it performs above 8.5 mA. So using just one I/O is still risky.
It might be safer to parallel two or three I/O pins, to ensure a strong-enough switch. Like this:



simulate this circuit &ndash; Schematic created using CircuitLab

Just be sure that those three I/O are always switched together high to de-activate the relay, low to activate the relay. Always with the same I2C command....you never want Out_1 fighting Out_2 or Out_3 with one pulling up while the others pull down.