Technically you cannot avoid the pdf content downloading to your device, your browser can't do anything nor display anything without downloading the actual content first. As far as viewing a PDF, it's residing in the browser's cache and you can either save it as file or just peruse through its contents via a tab but it has to be downloaded first. A browser just can't make up a PDF out of thin air on its own, it has to come from somewhere, in this case some online server.
That said, when viewing a web site with a PDF if you long press on the PDF link itself, Chrome should pop up a contextual menu including the options for 'Open in new tab' or 'Open in icognito tab. Is that not working for you?