Not really. The president of the United States has no formal power over the Supreme Court, besides nominating justices. They cannot tell the justices how to rule, nor can they "fire" them, so they have little power to compel obedience. The only recourse is impeachment, which requires the cooperation of Congress and has never happened to a Supreme Court justice since 1805, ironically in part for open partisanship. While justices can have political affiliations, it would be considered highly improper for a president to openly order a court to rule in their favor. That said, it might still happen. Similarly, federal judges have strict limits on their ability to be openly partisan, as noted in the previous link. Some judges—including Kavanaugh himself—do not even vote. Further, most Supreme Court rulings do not split along purely partisan lines (i.e. 5-4 in recent years):
The ratio is staggering. According to the Supreme Court Database, since 2000 a unanimous decision has been more likely than any other result — averaging 36 percent of all decisions. Even when the court did not reach a unanimous judgment, the justices often secured overwhelming majorities, with 7-to-2 or 8-to-1 judgments making up about 15 percent of decisions. The 5-to-4 decisions, by comparison, occurred in 19 percent of cases.
All this suggests that the two branches are still fairly separate.
However, much of the debate actually hinges on the political views of Kavanaugh (or any other judge), not their party affiliation per se. The worry is less that a judge will rubber-stamp anything their party tells them to, and more that their pre-existing political beliefs will influence their judicial rulings. For instance, consider the following excerpt from an email by DNC communications director Xochitl Hinojosa:
Even before these allegations surfaced, Kavanaugh's record was disqualifying. Here's what we know from his confirmation hearings:
• He failed to give a clear answer on his views on women's constitutional rights.
• He refused to promise to uphold the ACA's protections for people with pre-existing conditions.
• He dodged important questions on executive power.
If Kavanaugh is confirmed, Roe v. Wade, affordable health care, labor unions, and civil rights will all be on the chopping block. His confirmation would leave a stain on the Supreme Court and impact the direction of our country for decades to come -- long after Trump leaves the Oval Office.
Notice how the emphasis is less on the idea that Kavanaugh would do whatever Trump says, and more Kavanaugh's judicial views on reproductive health, Obamacare, and other issues that the Democratic party supports.
Similarly, many Republicans want justices who support what they consider the right opinions. While official statements are more neutral than campaign emails, naturally, talking about Kavanaugh's qualifications as a constitutionalist or a fair-minded judge, the prospect of Trump appointing conservative jurists was a factor in his support from the Republican establishment, and one that he played up, particularly in the context of abortion law:
"I am pro-life," Trump said during Wednesday night's presidential debate when asked whether he wanted that decision, Roe v. Wade, reversed by the Supreme Court.
Trump said that if the ruling were to be reversed, laws on the legality or illegality of abortion would "go back to the individual states" to decide, which was the case prior to Roe v. Wade.
But when moderator Chris Wallace pressed him on whether he wanted the ruling overturned, Trump said, "That will happen, automatically in my opinion," because he would get to nominate potentially several justices to the court.
In addition, judicial leanings more specific to a given political situation can come into play. One objection sometimes raised to Kavanaugh's nomination is his belief that the president cannot be subpoenaed, which he has been vague about in hearings. This is, of course, relevant to the investigation of Trump's campaign for alleged ties to the Russian government. As noted here:
The decision “took away the power of the president to control information in the executive branch by holding that the courts had power and jurisdiction to order the president to disclose information in response to a subpoena sought by a subordinate executive branch official,” Kavanaugh said.
“That was a huge step with implications to this day that most people do not appreciate sufficiently . . . Maybe the tension of the time led to an erroneous decision.” The discussion was first reported by the Associated Press.
Of course, Kavanaugh in particular now faces allegations of sexual assault, which provide an (ideally) non-partisan reason to object to his nomination as well.