In theory evaluating things in a bubble I think Roddick probably only wins about 4 slams without Federer. He gains 2 of Wimbledon 2003, Wimbledon 2004, Wimbledon 2009 if I had to guess, and probably (but not for sure) U.S Open 2006. Might have a shot at U.S Open 2007 but probably does not win. So adding to the U.S Open he already has that is about 4 slams.
I think all that seems reasonable, and pretty much in line with what I came up with.
However the X factor is if he does not fire Gilbert anymore, and I don't believe he does, how much does his career arc change, and how much is his confidence and momentum bolstered without Federer. In that scenario it isn't even impossible, as absurdly crazy as it sounds, that he might even win as much as 8 majors. Roddick IMO would be easily the worst and least talented player to win that much, but I could see a scenario that happens. I don't think he is ever truly dominant, but with the bolstered confidence and momentum he could win a slam each year from 2003-2009, maybe missing out in only 1 somewhere, and the odd year get 2 slams (maybe a Wimbledon- U.S Open double type thing) which could bring him to 8.
That's too much of a rabbit hole for me. Basically, that's supposing that Roddick improves to some degree rather than somewhat regressing during this period. That's a pretty big "if" regardless of who his coach is. This is along the lines of "what if Agassi got serious earlier" or "what if Capriati didn't have her issues" or "what if Sampras doesn't retire" or "what if Seles didn't get attacked." That's simply too big a leap for me with fantasy unknowns. After all, maybe if Andre gets serious earlier and plays with the mentality of veteran Andre, he wins 10-12 titles. It's easy to say, because it's constructed from amorphous ideas, there's no firm rationale to attack, no firm rationale to support it, either. Roddick's supposed Fed-less improvement with Gilbert that results in lots of slams apparently raises his tennis IQ, improves his backhand, causes him to volley and approach better and he's apparently more confident and has momentum. I'm not buying a ticket to this fantasy. Sorry. I bet a lot of players would win 8 or more slams if they had that magic makeover.
Frankly, lots of players regress from a brief peak, it's far more common that this happens than there is an improvement after a fairly high level - which is what 2003 was for Andy. Also common is when a good player has peaks and dips in his form and results over the course of years - you see this from players as good as Agassi and Becker, for example. Basically, Andy had this too, it's just that his peak level wasn't as high as those players. I'm not sure how much worse Andy was in 2004, it's just that Roger was improving and dominating to a degree that wasn't seen before.
However, removing Federer from the scene doesn't magically make Roddick a better player (aside from some Andy fantasy scenario) - he still has the same game with the strengths and weaknesses as before. It simply removes one of the greatest players ever from his competition. This doesn't just benefit Andy though, it opens the field for everyone during this period. Let's take a look and speculate a bit. I'm not employing any magical thinking, taking players pretty much at the level they actually played at. I'm also just looking at the field as a whole, not the specific draws at tournaments, since they wouldn't be the same minus Federer.
2003 - Minus Fed, Roddick is still in a bin with some others at the same approximate level, maybe a bit above. Agassi, Ferrero, Coria rounded out the top 5 with Moya and Nalbandian around as well. Hewitt seemed to have a down year in 2003, so I'm taking that into account even without Fed. Agassi's pretty good on the Hard Courts, winning AO, so obviously he's a contender at both the Australian and USO, as he was. Ferrero had quite a good year as well. Andy's not contending at the French anyway. Wimbledon is interesting, because while Roddick serve obviously plays well there, I think the speed of the court and the low bounce also expose some of the limitations of his game, aside from the serve. I think the hard courts are not quite as good for his serve (but close) but are slightly more forgiving of his weaknesses. So, while obviously Roddick has a good shot at Wimbledon in 2003, I think the absence of Fed also opens it up for the likes of Philippoussis, Henman, as well as the baseliners like Andre, Nalbandian, and even Ferrero in his best year. Those same guys, plus Hewitt and Moya and other baseliners would give Roddick competition at the USO. He's good in 2003, but Andy's not next level even in one of his best years. 1-2 slams is possible, I think 3 is pushing it.
2004 - Pretty similar story to 2004, but with Safin and Hewitt pushing their level up to Roddick's. This is a tougher field he's got to deal with this year with the addition of those two, particularly on the hard courts. Hewitt's a serious challenge at Wimbledon along the cast from 2003. I think the Andy has a shot at the AO but it's more likely to go to Andre, Safin, or Hewitt than him with Ferrero still playing well at this point and Nalbandian being a solid contender. Hewitt is an underrated grass court player and will provide better competition at Wimbledon this year than last. Roddick lost to Joachim Johansson in actual USO 04, hard to say about this alternate universe tournament. Andre and Hewitt would be serious contenders. I think Andy wins a slam, maybe 2, but not more.
2005 - Again similar to 2004. I think peak Safin wins at AO either in 2004 or this year (or both). Wimbledon is the same cast, with Hewitt contending with Roddick. Does Thomas Johansson, Gonzales, Nalbandian have an opening this year? Hmm... I think Andre wins the USO without peak Fed to tangle with, a fitting slam win to book end his career. Minus Fed, I think Andre get 1-2 more slams (maybe a 3rd, maybe). It's not just Roddick that would benefit, I think Hewitt gets to 3 or maybe 4, Safin to 3 or 4. These slams would get sprinkled among these players, as none of them are dominant - though all are good or great. Frankly, it's pretty likely that someone somewhat random would pick up a their only slam win, maybe Nalbandian or Henman or Philippoussis during this period, somewhere. Maybe 1 slam for Andy.
2006 - Things start to change here. Safin disappears and Hewitt is injured. Things might look rosy for Roddick, except this Spanish kid has started to emerge. Andy didn't fare well in AO this year, so no love there. Wimbledon is always good for him, but there's this Nadal guy. He's not great on the grass, yet, but pretty good. Maybe a Nadal win, maybe someone else like Nieminen, or Bjorkman or Baghdatis or Hewitt or even Stepanek for you serve and volleyers. Andy has a shot, but I'll take the field over him, here. At the USO, beast-mode Blake could win or Hewitt, or Rafa, or even Davydenko, but Roddick's got a good shot. I'll take him slightly over the field. Slightly. Maybe 1 slam for Andy.
2007 - Now there are more issues. This Nadal guy keeps getting better. This Novak teenager shows up. Roddick has a decent shot at what is likely his first AO. Maybe Rafa, gets on the board there, early, though. Nobody is beating Rafa at either Wimbledon or the French. Novak probably wins the USO.
2008 - There's this Murray guy that starts winning more. Novak wins the AO. Rafa the grass and clay. Maybe Murray wins the USO (or Novak or Rafa). Now Andy's got 3 guys who are better than him to contend with regularly. Uh oh.
2009 - AO - Rafa (or Novak). Roddick's not winning in Roland Garros, so even if Rafa falls to Soderling in this alternate universe, he doesn't benefit. If Rafa's hurt at Wimbledon, the Andy could win it - or Murray or Novak. If Rafa's healthy I think it's 3 in a row. USO is Delpo, Murray, Novak, (or Rafa or someone random)
It doesn't get better from here.
So, while Andy obviously benefits from the absence of Roger, I think lots of people do. I think Agassi, Hewitt, Safin, and someone else all pick (random) all pick up slams in Fed's absence in the early years 2003-2005. From 2006 on, Rafa, Novak and Murray pick up most of those slams. Frankly, they are all better players, more complete players. As I mention before, while Wimbledon gives Roddick's serve a boost, it also exposes his tactics, his mediocre approaches and volleys and movement. There are people that think that Andy can serve his way to many Wimbledon titles minus Roger. I don't think so. I think it's likely he wins at least one. However, nobody's beating Rafa at Wimbledon in 2007 or 2008 except peak Roger.
There's no magic way to get Andy to 8 or more slams (maybe some kind of fairy magic) because during his peak, there were players essentially at the same level as him. Then, starting around 2006 there are players that are flat out better than him (like Roger was).