Well, they can fight as much as they like, but at the end of the day, the Republicans have enough votes to confirm Gorsuch as soon as McConnell chooses to go for the so-called "nuclear option" rule change. He's one who stills puts on the masquerade of "the comity of the senate" so he'll wait awhile before going ahead, but he'll do it if the occasion requires it. Gorsuch by all accounts is a pretty staunch conservative but I haven't yet seen any red flags that would disqualify him like Harriet Miers, etc.
As for the other justices and seats, the only question mark is Justice Ginsburg really, sorry to be so morbid. It's obviously a difficult decision for anyone who loves his or her work to retire, and she still very much enjoys working apparently, but she has potentially shifted the power balance of the court for the next decade if anything happens to her.
The level of fight will be dictated by the selection of the judge in question. If Trump picks someone nutty for the next opening (if one comes about) like say, Ted Cruz, he may face a real opposition. I just don't see the Gorsuch choice lending itself itself as much to public controversy