1. There is no need of using any server side processing. One of the modern web app development concepts is front-end and back-end separation. If you can do it on front-end, then don't use server to do the job.
2. conditionalPanel is a better solution but users can still click the tab button, just give them an empty page.

Here is an even better solution, let's use some js to disable the tab button unless users click the action button. Users can see the tab button but it's gray and unclickable on start:

library(shiny)

ui <- shinyUI(navbarPage(
  "",
  tabPanel(
    h1("Tab1"), 
    value = "nav1",
    mainPanel(
      br(),
      h2("The user must press this button to access the other tab."),
      br(),
      shiny::actionButton('button', 'press the button', onclick = "$(tab).removeClass('disabled')")
    )
  ),
  tabPanel(
    h1("Tab2"),
    value = "nav2",
    uiOutput("tab2contents")
  ),
  tags$script(
    '
    var tab = $(\'a[data-value="nav2"]\').parent().addClass("disabled");
    $(function(){
      $(tab.parent()).on("click", "li.disabled", function(e) {
        e.preventDefault();
        return false;
      });
    });
    '
  )
))    

server <- shinyServer(function(input, output) {

})

# Run the application
shinyApp(ui = ui, server = server)

Use conditionalPanel(). Condition? The button shouldn't have zero clicks.

Your example now becomes:

library(shiny)

ui <- shinyUI(
  navbarPage(
    title = "", 
    
    tabPanel(
      title = h1("Tab1"), 
      value = "nav1",
      
      mainPanel(
        br(),
        h2("The user must press this button to access the other tab."),
        br(),
        shiny::actionButton('button', 'press the button')
      )
    ), 
    
    tabPanel(
      h1("Tab2"),
      value = "nav2",
      
      # ----conditional panel here----
      conditionalPanel(
        condition = "input.button != 0", 
        
        h3('Block access until user presses button')
      )
    )
  )
)

server <- shinyServer(function(input, output) {
  
  
})

# Run the application
shinyApp(ui = ui, server = server)

Adding detail to my comment above:

library(shiny)

ui <- shinyUI(navbarPage("",
        tabPanel(
          h1("Tab1"), 
          value = "nav1",
          mainPanel(
            br(),
            h2("The user must press this button to access the other tab."),
            br(),
            shiny::actionButton('button', 'press the button')
          )
        ),
        tabPanel(
          h1("Tab2"),
          value = "nav2",
          uiOutput("tab2contents")
        )
      )
    )    

server <- shinyServer(function(input, output) {
   v <- reactiveValues(tab2Active=FALSE)
   
  observeEvent(input$button, { v$tab2Active <- TRUE})

  output$tab2contents <- renderUI({
    if (v$tab2Active) {
      h3('Tab 2 is active')
    } else {
      h3('Block access until user presses button')
    }
  })
})

# Run the application
shinyApp(ui = ui, server = server)