Yes of course you can publish from Power BI Desktop to Power BI Service without a gateway.
Do you have an Office 365 account ?
Simply publish your report in your "personal workspace" or in an Office 365 group/workspace if you have a pro version.
Then open your report in Power BI Service (App.powerbi.com).
The gateway will be helpful only if you want to keep your data up to date without having to go back to Power BI Desktop, refresh and re-publish your report.
It basically makes the link between on-premise and cloud environment and automatically refreshes the data.