Can a board like Arduino charge a Li-ion 3.7V battery?
Dont use the Arduino to charge the battery , you need a way to monitor the voltage levels when charging and restrict the amount of current based on state of charge.
It would be probably safer to use a 3.3v power supply and pulse that on and off with the Arduino , than say, use the PWM output with a voltage divider to lower the voltage across the battery and charge that way.
Either suggestion from above is not ideal and not recommended