Can I keep Li-Ion battery 'charging' continuously at 3.7 V?
does the battery take zero current once it's at 3.7 V?


Yes. Or to be more precise, once the battery voltage reaches the regulated power supply voltage the charge current will reduce exponentially until it reaches the battery's leakage current (which is very low). 


  Can I keep Li-Ion battery 'charging' continuously at 3.7 V?


Yes, you can do that. But the battery will only be charged to ~10% of its rated capacity. 

Apart from the obvious capacity issue, this could be a big problem if the power supply input is removed and the battery is not switched out of circuit soon after. If the battery drops below 2.5 V it will be permanently damaged and won't charge. When you reconnect power the LM2596 will try to charge it with up to 3 A of current, which could damage it if the voltage is below 3.0 V. Trying to charge a damaged Li-ion battery at high current may cause it to explode! 

Bottom line: for safety and to prevent damaging the battery you should use a proper li-ion charger which is designed to 'float' the battery while powering the load. Here's an example:-

2.5A Single Cell Switch Mode Battery Charger with Power Path Management