Virtually all oil is API certified. The question is the proper weight. Your Honda manual states 0W-20. It does not give any alternate weights. If you want to use 10w30, or 10w40 or whatever viscosity you want, you take the risk of being denied warranty coverage should you develop an oil related engine problem.
It is very common for shops or vehicle inspectors to send an oil sample to a lab for analysis.
Personally, and similar to the thought on many Honda boards is that the 0w20 spec is too thin and was only choosen by Honda to get that slightly better gas mileage. There has been a high coincidence of engines using 0w20 burning too much oil in the 10 to 20 thousand mile range. Many are saying to move to the 5w20 Mobil 1 instead. Obviously, if you want to deviate from anything in your owners manual, you do so at your own risk.