Can I use 12 V 7 Ah battery with 700 VA inverter?
Unfortunately you haven't provided enough information to answer the question fully. What you need to know is the discharge current characteristics for the battery and the specifications for the inverter.

A 60W equivalent CFL is likely to be somewhere in the region of 15W. So, your load will be somewhere in the region of 30W.

If your inverter is rated for 700VA then you can think of it as being able to support a 700W load (for the real, and longer, answer search for volt-amps). Therefore the load on your inverter would be well within its capabilities at less than 10%. A smaller inverter might suit your needs better.

If this is a cheap Chinese square-wave inverter then the efficiency could be as poor as 50%.

So: 30W / 0.5 = 60W input to the inverter. Or: 60W / 12V = 5A

Usually even SLA batteries can support a 1C load (7A in your case for a 7AH battery), so you would be comfortably under that.

BUT, inverters can have significant inrush current. The start-up characteristics should be in the documentation for the inverter, you just need to check two things on the battery spec:


That the battery can support this short-term spike in current without damage while the inverter starts up. 
That the terminal voltage will not fall below the under-voltage cutout for the inverter due to the battery's internal resistance.