Ultimately, Derrick White will make an impact on Walker's future
While Walker simply outplaying White could get him a starting job, there are still other ways in which it could happen. The other two most likely scenarios that could play out both happen to hinge on what happens with White this season.
One possibility is that White is handed the responsibility of leading the bench unit as their point guard. Because he already has experience as the team's starting point guard (when Dejounte Murray sat with a torn ACL), this is a possibility that we've discussed here before. White would essentially provide the bench unit with a more consistent and versatile skillset to build upon compared to Walker, while Walker could provide the already defensive-heavy starting lineup with a dose of scoring.
On another note, another very real possibility (as unpleasant as it may be) is that White finds himself injured again this season. Outside of the NBA's health and safety protocols that kept him out of several games last year, White was held back from a few nagging injuries including an ankle injury that ultimately ended his season.
As we've noted in the past, White is by far the oldest of the Spurs' more recent draft picks at 27. Additionally, while he hasn't had any career-altering injuries up to this point, he's had several nagging injuries focused in his lower body, particularly in his feet.
As good of a player as White is, if he's unable to remain healthy and properly conditioned throughout the season, this is something that could quickly propel Walker into the starting lineup.
Regardless of White, though, the quickest and most effective way for Walker to earn a starting spot will be for him to show that he can make his presence felt in every game that he plays. He has the elite athleticism to outrun and out-jump opponents, and he has the skills to out-finesse opponents, but this coming season will be about him putting those things together to tap into his potential. Personally, I have no doubt that he can do it.