No, there is not even a good correlation between actual rape rates and immigration.
The figure that Gatestone Institute cites is the rate of reported rapes. This has indeed increased, but the reason has nothing to do with a supposed increase in actual rapes.
This is because crime victim surveys have shown that the actual rates of rape have remained relatively unchanged from 2005 (when measurements started) to 2014 (latest year for which summary statistics is available). So we know that actual rates of rape are more or less constant during that period and violent crime has decreased overall for decades (according to The Better Angels of Our Nature: Why Violence Has Declined by Steven Pinker).
So why have the rate of reported rapes increased? There are a few reasons:
Expansion of rape definition in several steps: actions that are comparable to rape (1998), helpless state (2005) and particularly vulnerable situation (2013) .
Increased tendency to report crime: the observed tendency to report crime, as measured by number of rape reports / number of rapes in crime victim surveys has doubled between 2005 (10%) and 2010 (20%) .
Changes in how police handle rape reports: Swedish police makes one police report per rape, so if a person has been raped 30 times in a relationship, it will count as 30 separate rape crimes and they have made a conscious effort to file all sex crimes that could be rape as rape even though they might end up as being another sex crime or no crime at all .
So there is not even a correlation to speak of. Sweden has had lots of immigration during 2005-2014, yet more or less constant actual rates of rape.
Anti-immigration activists often retort by claiming that immigrants are overrepresented in crime statistics and while true, the observed overrepresentation is smaller for immigrants (2.5x) than for men (3.5x) and people who are unemployed, on welfare or without high school education (5x-6x), so they are blowing it way out of proportion .
All references (except the last one) come from the Swedish Council for Crime Prevention (Brottsförebyggande Rådet, BRÅ), which is a Swedish governmental agency that collects statistics and writes reports about crime. The last reference is from the National Centre for Knowledge on Men's Violence Against Women, University of Uppsala.
Unfortunately, they are almost all in Swedish (since they are primary sources), but BRÅ is a recognized authority on crime statistics in Sweden, even among the anti-immigration activists.
 https://web.archive.org/web/20170228040947/https://www.bra.se/bra/brott-och-statistik/statistik/utsatthet-for-brott/ntu.html (red line is sex crimes, yearly reports contain information about rapes specifically)
 https://www.bra.se/download/18.cba82f7130f475a2f1800012829/2005_07_valdtakt_kartlaggning.pdf (p. 16)
 https://www.bra.se/download/18.744c0a913040e4033180001042/2011_6_polisanmalda_valdtkter_barn.pdf (pp. 9-10)
 https://www.bra.se/download/18.cba82f7130f475a2f1800012697/2005_17_brottslighet_bland_personer_fodda_sverige_och_utlandet.pdf (p. 35 )