I know if I buy it at Gatwick airport I can but If I don't know if I do at Downtown, will it work? My idea is to buy one at Heathrow and then go to visit London and take my connection to Gatwick. Am I right? What do you recommend me? If I pay by contactless CC, how secure and cheaper is compared to get a one day unlimited transport card?
If you pay by contactless, do not worry about travelcards. The fare you pay will be automatically limited by the daily fare cap. The cap used to be the same as the price of the day travelcard that would have covered the same journeys, but is now generally even less than that.
(Beware that Heathrow Express and Gatwick Express charge special fares that do not count towards the cap. But nor are travelcards valid for them anyway. Traveling to/from the airport by tube, TfL Rail, or regular Southern/Thameslink trains do count for the cap).
Since nobody actually answered your question as phrased, I thought I'd make this very clear:
No, a Zone 1-9 Travelcard is NOT valid to Gatwick Airport.
You can see a map of Travelcard validity on the London's Rail and Tube Services map.
You can see that while Gatwick Airport is in the Oyster Pay as you Go area (meaning you can use Oyster or contactless bank cards to pay for travel), it does NOT have a zone number listed, meaning it is not valid with ordinary numbered-zone Travelcards.
You can still buy a Travelcard from Gatwick Airport if you wish but it's a so-called "outboundary" Travelcard, which is in effect just a one-day return (Americans would say "round trip") ticket to the boundary of Zone 6, combined with a Travelcard valid within Zones 1-6. There are various available depending on whether you need validity on Gatwick Express trains or not and whether or not you're travelling at peak times, but in any case, it will almost certainly work out more expensive than just using Oyster or Contactless. Henning gave more details on how this works in his answer.